:: QUATERNI semantic presentation
:: deftheorem Def1 defines QUATERNION QUATERNI:def 1 :
:: deftheorem Def2 defines quaternion QUATERNI:def 2 :
definition
let x be
set ;
let y be
set ;
let w be
set ;
let z be
set ;
let a be
set ;
let b be
set ;
let c be
set ;
let d be
set ;
func c1,
c2,
c3,
c4 --> c5,
c6,
c7,
c8 -> set equals :: QUATERNI:def 3
(x,y --> a,b) +* (w,z --> c,d);
coherence
(x,y --> a,b) +* (w,z --> c,d) is set
;
end;
:: deftheorem Def3 defines --> QUATERNI:def 3 :
for
x,
y,
w,
z,
a,
b,
c,
d being
set holds
x,
y,
w,
z --> a,
b,
c,
d = (x,y --> a,b) +* (w,z --> c,d);
registration
let x be
set ;
let y be
set ;
let w be
set ;
let z be
set ;
let a be
set ;
let b be
set ;
let c be
set ;
let d be
set ;
cluster a1,
a2,
a3,
a4 --> a5,
a6,
a7,
a8 -> Relation-like Function-like ;
coherence
( x,y,w,z --> a,b,c,d is Function-like & x,y,w,z --> a,b,c,d is Relation-like )
;
end;
theorem Th1: :: QUATERNI:1
for
x,
y,
w,
z,
a,
b,
c,
d being
set holds
dom (x,y,w,z --> a,b,c,d) = {x,y,w,z}
theorem Th2: :: QUATERNI:2
for
x,
y,
w,
z,
a,
b,
c,
d being
set holds
rng (x,y,w,z --> a,b,c,d) c= {a,b,c,d}
Lemma38:
0,1,2,3 are_mutually_different
by INCPROJ:def 6;
theorem Th3: :: QUATERNI:3
for
x,
y,
w,
z,
a,
b,
c,
d being
set st
x,
y,
w,
z are_mutually_different holds
(
(x,y,w,z --> a,b,c,d) . x = a &
(x,y,w,z --> a,b,c,d) . y = b &
(x,y,w,z --> a,b,c,d) . w = c &
(x,y,w,z --> a,b,c,d) . z = d )
Lemma45:
for x, y, w, z, a, b, c, d being set st x,y,w,z are_mutually_different holds
{a,b,c,d} c= rng (x,y,w,z --> a,b,c,d)
theorem Th4: :: QUATERNI:4
for
x,
y,
w,
z,
a,
b,
c,
d being
set st
x,
y,
w,
z are_mutually_different holds
rng (x,y,w,z --> a,b,c,d) = {a,b,c,d}
theorem Th5: :: QUATERNI:5
for
x1,
x2,
x3,
x4,
X being
set holds
(
{x1,x2,x3,x4} c= X iff (
x1 in X &
x2 in X &
x3 in X &
x4 in X ) )
definition
let A be non
empty set ;
let x be
set ;
let y be
set ;
let w be
set ;
let z be
set ;
let a be
Element of
A,
b be
Element of
A,
c be
Element of
A,
d be
Element of
A;
redefine func --> as
c2,
c3,
c4,
c5 --> c6,
c7,
c8,
c9 -> Function of
{a2,a3,a4,a5},
a1;
coherence
x,y,w,z --> a,b,c,d is Function of {x,y,w,z},A
end;
definition
func <j> -> set equals :: QUATERNI:def 4
0,1,2,3
--> 0,0,1,0;
coherence
0,1,2,3 --> 0,0,1,0 is set
;
func <k> -> set equals :: QUATERNI:def 5
0,1,2,3
--> 0,0,0,1;
coherence
0,1,2,3 --> 0,0,0,1 is set
;
end;
:: deftheorem Def4 defines <j> QUATERNI:def 4 :
:: deftheorem Def5 defines <k> QUATERNI:def 5 :
definition
let x be
Element of
REAL ,
y be
Element of
REAL ,
w be
Element of
REAL ,
z be
Element of
REAL ;
func [*c1,c2,c3,c4*] -> Element of
QUATERNION equals :
Def6:
:: QUATERNI:def 6
[*x,y*] if (
w = 0 &
z = 0 )
otherwise 0,1,2,3
--> x,
y,
w,
z;
consistency
for b1 being Element of QUATERNION holds verum
;
coherence
( ( w = 0 & z = 0 implies [*x,y*] is Element of QUATERNION ) & ( ( not w = 0 or not z = 0 ) implies 0,1,2,3 --> x,y,w,z is Element of QUATERNION ) )
end;
:: deftheorem Def6 defines [* QUATERNI:def 6 :
for
x,
y,
w,
z being
Element of
REAL holds
( (
w = 0 &
z = 0 implies
[*x,y,w,z*] = [*x,y*] ) & ( ( not
w = 0 or not
z = 0 ) implies
[*x,y,w,z*] = 0,1,2,3
--> x,
y,
w,
z ) );
theorem Th6: :: QUATERNI:6
for
a,
b,
c,
d,
e,
i,
j,
k being
set for
g being
Function st
a <> b &
c <> d &
dom g = {a,b,c,d} &
g . a = e &
g . b = i &
g . c = j &
g . d = k holds
g = a,
b,
c,
d --> e,
i,
j,
k
theorem Th7: :: QUATERNI:7
theorem Th8: :: QUATERNI:8
for
a,
c,
x,
w,
b,
d,
y,
z being
set st
a,
c,
x,
w are_mutually_different holds
a,
c,
x,
w --> b,
d,
y,
z = {[a,b],[c,d],[x,y],[w,z]}
Lemma61:
for x, y, z being set st [x,y] = {z} holds
( z = {x} & x = y )
theorem Th9: :: QUATERNI:9
Lemma75:
for a, b, c, d being Element of REAL holds not 0,1,2,3 --> a,b,c,d in REAL
theorem Th10: :: QUATERNI:10
theorem Th11: :: QUATERNI:11
for
a,
b,
c,
d,
x,
y,
z,
w,
x',
y',
z',
w' being
set st
a,
b,
c,
d are_mutually_different &
a,
b,
c,
d --> x,
y,
z,
w = a,
b,
c,
d --> x',
y',
z',
w' holds
(
x = x' &
y = y' &
z = z' &
w = w' )
theorem Th12: :: QUATERNI:12
for
x1,
x2,
x3,
x4,
y1,
y2,
y3,
y4 being
Element of
REAL st
[*x1,x2,x3,x4*] = [*y1,y2,y3,y4*] holds
(
x1 = y1 &
x2 = y2 &
x3 = y3 &
x4 = y4 )
definition
let x be
quaternion number ,
y be
quaternion number ;
x in QUATERNION
by ;
then consider x1 being
Element of
REAL ,
x2 being
Element of
REAL ,
x3 being
Element of
REAL ,
x4 being
Element of
REAL such that E37:
x = [*x1,x2,x3,x4*]
by ;
y in QUATERNION
by ;
then consider y1 being
Element of
REAL ,
y2 being
Element of
REAL ,
y3 being
Element of
REAL ,
y4 being
Element of
REAL such that E40:
y = [*y1,y2,y3,y4*]
by ;
func c1 + c2 -> set means :
Def7:
:: QUATERNI:def 7
ex
x1,
x2,
x3,
x4,
y1,
y2,
y3,
y4 being
Element of
REAL st
(
x = [*x1,x2,x3,x4*] &
y = [*y1,y2,y3,y4*] &
it = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] );
existence
ex b1 being set ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] )
uniqueness
for b1, b2 being set st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) & ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b2 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) holds
b1 = b2
commutativity
for b1 being set
for x, y being quaternion number st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) holds
ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( y = [*x1,x2,x3,x4*] & x = [*y1,y2,y3,y4*] & b1 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] )
;
end;
:: deftheorem Def7 defines + QUATERNI:def 7 :
for
x,
y being
quaternion number for
b3 being
set holds
(
b3 = x + y iff ex
x1,
x2,
x3,
x4,
y1,
y2,
y3,
y4 being
Element of
REAL st
(
x = [*x1,x2,x3,x4*] &
y = [*y1,y2,y3,y4*] &
b3 = [*(x1 + y1),(x2 + y2),(x3 + y3),(x4 + y4)*] ) );
Lemma139:
0 = [*0,0,0,0*]
:: deftheorem Def8 defines - QUATERNI:def 8 :
:: deftheorem Def9 defines - QUATERNI:def 9 :
definition
let x be
quaternion number ,
y be
quaternion number ;
x in QUATERNION
by ;
then consider x1 being
Element of
REAL ,
x2 being
Element of
REAL ,
x3 being
Element of
REAL ,
x4 being
Element of
REAL such that E37:
x = [*x1,x2,x3,x4*]
by ;
y in QUATERNION
by ;
then consider y1 being
Element of
REAL ,
y2 being
Element of
REAL ,
y3 being
Element of
REAL ,
y4 being
Element of
REAL such that E40:
y = [*y1,y2,y3,y4*]
by ;
func c1 * c2 -> set means :
Def10:
:: QUATERNI:def 10
ex
x1,
x2,
x3,
x4,
y1,
y2,
y3,
y4 being
Element of
REAL st
(
x = [*x1,x2,x3,x4*] &
y = [*y1,y2,y3,y4*] &
it = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] );
existence
ex b1 being set ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] )
uniqueness
for b1, b2 being set st ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b1 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) & ex x1, x2, x3, x4, y1, y2, y3, y4 being Element of REAL st
( x = [*x1,x2,x3,x4*] & y = [*y1,y2,y3,y4*] & b2 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) holds
b1 = b2
end;
:: deftheorem Def10 defines * QUATERNI:def 10 :
for
x,
y being
quaternion number for
b3 being
set holds
(
b3 = x * y iff ex
x1,
x2,
x3,
x4,
y1,
y2,
y3,
y4 being
Element of
REAL st
(
x = [*x1,x2,x3,x4*] &
y = [*y1,y2,y3,y4*] &
b3 = [*((((x1 * y1) - (x2 * y2)) - (x3 * y3)) - (x4 * y4)),((((x1 * y2) + (x2 * y1)) + (x3 * y4)) - (x4 * y3)),((((x1 * y3) + (y1 * x3)) + (y2 * x4)) - (y4 * x2)),((((x1 * y4) + (x4 * y1)) + (x2 * y3)) - (x3 * y2))*] ) );
definition
redefine func <j> as
<j> -> Element of
QUATERNION equals :: QUATERNI:def 11
[*0,0,1,0*];
coherence
<j> is Element of QUATERNION
by ;
compatibility
for b1 being Element of QUATERNION holds
( b1 = <j> iff b1 = [*0,0,1,0*] )
by ;
redefine func <k> as
<k> -> Element of
QUATERNION equals :: QUATERNI:def 12
[*0,0,0,1*];
coherence
<k> is Element of QUATERNION
by ;
compatibility
for b1 being Element of QUATERNION holds
( b1 = <k> iff b1 = [*0,0,0,1*] )
by ;
end;
:: deftheorem Def11 defines <j> QUATERNI:def 11 :
:: deftheorem Def12 defines <k> QUATERNI:def 12 :
theorem Th13: :: QUATERNI:13
theorem Th14: :: QUATERNI:14
theorem Th15: :: QUATERNI:15
theorem Th16: :: QUATERNI:16
theorem Th17: :: QUATERNI:17
theorem Th18: :: QUATERNI:18
theorem Th19: :: QUATERNI:19
theorem Th20: :: QUATERNI:20
theorem Th21: :: QUATERNI:21
definition
let z be
quaternion number ;
func Rea c1 -> set means :
Def13:
:: QUATERNI:def 13
ex
z' being
complex number st
(
z = z' &
it = Re z' )
if z in COMPLEX otherwise ex
f being
Function of 4,
REAL st
(
z = f &
it = f . 0 );
existence
( ( z in COMPLEX implies ex b1 being set ex z' being complex number st
( z = z' & b1 = Re z' ) ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4, REAL st
( z = f & b1 = f . 0 ) ) )
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & ex z' being complex number st
( z = z' & b1 = Re z' ) & ex z' being complex number st
( z = z' & b2 = Re z' ) implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4, REAL st
( z = f & b1 = f . 0 ) & ex f being Function of 4, REAL st
( z = f & b2 = f . 0 ) implies b1 = b2 ) )
;
consistency
for b1 being set holds verum
;
func Im1 c1 -> set means :
Def14:
:: QUATERNI:def 14
ex
z' being
complex number st
(
z = z' &
it = Im z' )
if z in COMPLEX otherwise ex
f being
Function of 4,
REAL st
(
z = f &
it = f . 1 );
existence
( ( z in COMPLEX implies ex b1 being set ex z' being complex number st
( z = z' & b1 = Im z' ) ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4, REAL st
( z = f & b1 = f . 1 ) ) )
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & ex z' being complex number st
( z = z' & b1 = Im z' ) & ex z' being complex number st
( z = z' & b2 = Im z' ) implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4, REAL st
( z = f & b1 = f . 1 ) & ex f being Function of 4, REAL st
( z = f & b2 = f . 1 ) implies b1 = b2 ) )
;
consistency
for b1 being set holds verum
;
func Im2 c1 -> set means :
Def15:
:: QUATERNI:def 15
it = 0
if z in COMPLEX otherwise ex
f being
Function of 4,
REAL st
(
z = f &
it = f . 2 );
existence
( ( z in COMPLEX implies ex b1 being set st b1 = 0 ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4, REAL st
( z = f & b1 = f . 2 ) ) )
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4, REAL st
( z = f & b1 = f . 2 ) & ex f being Function of 4, REAL st
( z = f & b2 = f . 2 ) implies b1 = b2 ) )
;
consistency
for b1 being set holds verum
;
func Im3 c1 -> set means :
Def16:
:: QUATERNI:def 16
it = 0
if z in COMPLEX otherwise ex
f being
Function of 4,
REAL st
(
z = f &
it = f . 3 );
existence
( ( z in COMPLEX implies ex b1 being set st b1 = 0 ) & ( not z in COMPLEX implies ex b1 being set ex f being Function of 4, REAL st
( z = f & b1 = f . 3 ) ) )
uniqueness
for b1, b2 being set holds
( ( z in COMPLEX & b1 = 0 & b2 = 0 implies b1 = b2 ) & ( not z in COMPLEX & ex f being Function of 4, REAL st
( z = f & b1 = f . 3 ) & ex f being Function of 4, REAL st
( z = f & b2 = f . 3 ) implies b1 = b2 ) )
;
consistency
for b1 being set holds verum
;
end;
:: deftheorem Def13 defines Rea QUATERNI:def 13 :
:: deftheorem Def14 defines Im1 QUATERNI:def 14 :
:: deftheorem Def15 defines Im2 QUATERNI:def 15 :
:: deftheorem Def16 defines Im3 QUATERNI:def 16 :
theorem Th22: :: QUATERNI:22
for
f being
Function of 4,
REAL ex
a,
b,
c,
d being
Element of
REAL st
f = 0,1,2,3
--> a,
b,
c,
d
Lemma149:
for a, b being Element of REAL holds
( Re [*a,b*] = a & Im [*a,b*] = b )
Lemma152:
for z being complex number holds [*(Re z),(Im z)*] = z
theorem Th23: :: QUATERNI:23
for
a,
b,
c,
d being
Element of
REAL holds
(
Rea [*a,b,c,d*] = a &
Im1 [*a,b,c,d*] = b &
Im2 [*a,b,c,d*] = c &
Im3 [*a,b,c,d*] = d )
theorem Th24: :: QUATERNI:24
theorem Th25: :: QUATERNI:25
:: deftheorem Def17 defines 0q QUATERNI:def 17 :
:: deftheorem Def18 defines 1q QUATERNI:def 18 :
Lemma162:
for a, b, c, d being real number st (((a ^2 ) + (b ^2 )) + (c ^2 )) + (d ^2 ) = 0 holds
( a = 0 & b = 0 & c = 0 & d = 0 )
theorem Th26: :: QUATERNI:26
theorem Th27: :: QUATERNI:27
theorem Th28: :: QUATERNI:28
Lemma164:
[*1,0,0,0*] = 1
theorem Th29: :: QUATERNI:29
theorem Th30: :: QUATERNI:30
theorem Th31: :: QUATERNI:31
Lemma167:
for m, n, x, y, z being quaternion number st z = ((m + n) + x) + y holds
( Rea z = (((Rea m) + (Rea n)) + (Rea x)) + (Rea y) & Im1 z = (((Im1 m) + (Im1 n)) + (Im1 x)) + (Im1 y) & Im2 z = (((Im2 m) + (Im2 n)) + (Im2 x)) + (Im2 y) & Im3 z = (((Im3 m) + (Im3 n)) + (Im3 x)) + (Im3 y) )
Lemma176:
for x, y, z being quaternion number st z = x + y holds
( Rea z = (Rea x) + (Rea y) & Im1 z = (Im1 x) + (Im1 y) & Im2 z = (Im2 x) + (Im2 y) & Im3 z = (Im3 x) + (Im3 y) )
Lemma177:
for z1, z2 being quaternion number holds z1 + z2 = [*((Rea z1) + (Rea z2)),((Im1 z1) + (Im1 z2)),((Im2 z1) + (Im2 z2)),((Im3 z1) + (Im3 z2))*]
Lemma178:
for x, y, z being quaternion number st z = x * y holds
( Rea z = ((((Rea x) * (Rea y)) - ((Im1 x) * (Im1 y))) - ((Im2 x) * (Im2 y))) - ((Im3 x) * (Im3 y)) & Im1 z = ((((Rea x) * (Im1 y)) + ((Im1 x) * (Rea y))) + ((Im2 x) * (Im3 y))) - ((Im3 x) * (Im2 y)) & Im2 z = ((((Rea x) * (Im2 y)) + ((Im2 x) * (Rea y))) + ((Im3 x) * (Im1 y))) - ((Im1 x) * (Im3 y)) & Im3 z = ((((Rea x) * (Im3 y)) + ((Im3 x) * (Rea y))) + ((Im1 x) * (Im2 y))) - ((Im2 x) * (Im1 y)) )
Lemma179:
for z1, z2, z3, z4 being quaternion number holds ((z1 + z2) + z3) + z4 = [*((((Rea z1) + (Rea z2)) + (Rea z3)) + (Rea z4)),((((Im1 z1) + (Im1 z2)) + (Im1 z3)) + (Im1 z4)),((((Im2 z1) + (Im2 z2)) + (Im2 z3)) + (Im2 z4)),((((Im3 z1) + (Im3 z2)) + (Im3 z3)) + (Im3 z4))*]
Lemma180:
for z1, z2 being quaternion number holds z1 * z2 = [*(((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2))),(((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2))),(((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2))),(((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2)))*]
Lemma181:
for z1, z2 being quaternion number holds
( Rea (z1 * z2) = ((((Rea z1) * (Rea z2)) - ((Im1 z1) * (Im1 z2))) - ((Im2 z1) * (Im2 z2))) - ((Im3 z1) * (Im3 z2)) & Im1 (z1 * z2) = ((((Rea z1) * (Im1 z2)) + ((Im1 z1) * (Rea z2))) + ((Im2 z1) * (Im3 z2))) - ((Im3 z1) * (Im2 z2)) & Im2 (z1 * z2) = ((((Rea z1) * (Im2 z2)) + ((Im2 z1) * (Rea z2))) + ((Im3 z1) * (Im1 z2))) - ((Im1 z1) * (Im3 z2)) & Im3 (z1 * z2) = ((((Rea z1) * (Im3 z2)) + ((Im3 z1) * (Rea z2))) + ((Im1 z1) * (Im2 z2))) - ((Im2 z1) * (Im1 z2)) )
theorem Th32: :: QUATERNI:32
Lemma182:
for z being quaternion number
for x being Real st z = x holds
( Rea z = x & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 )
theorem Th33: :: QUATERNI:33
theorem Th34: :: QUATERNI:34
theorem Th35: :: QUATERNI:35
definition
let x be
Real;
let y be
quaternion number ;
y in QUATERNION
by ;
then consider y1 being
Element of
REAL ,
y2 being
Element of
REAL ,
y3 being
Element of
REAL ,
y4 being
Element of
REAL such that E37:
y = [*y1,y2,y3,y4*]
by ;
func c1 + c2 -> set means :
Def19:
:: QUATERNI:def 19
ex
y1,
y2,
y3,
y4 being
Element of
REAL st
(
y = [*y1,y2,y3,y4*] &
it = [*(x + y1),y2,y3,y4*] );
existence
ex b1 being set ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x + y1),y2,y3,y4*] )
uniqueness
for b1, b2 being set st ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x + y1),y2,y3,y4*] ) & ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b2 = [*(x + y1),y2,y3,y4*] ) holds
b1 = b2
end;
:: deftheorem Def19 defines + QUATERNI:def 19 :
:: deftheorem Def20 defines - QUATERNI:def 20 :
definition
let x be
Real;
let y be
quaternion number ;
y in QUATERNION
by ;
then consider y1 being
Element of
REAL ,
y2 being
Element of
REAL ,
y3 being
Element of
REAL ,
y4 being
Element of
REAL such that E37:
y = [*y1,y2,y3,y4*]
by ;
func c1 * c2 -> set means :
Def21:
:: QUATERNI:def 21
ex
y1,
y2,
y3,
y4 being
Element of
REAL st
(
y = [*y1,y2,y3,y4*] &
it = [*(x * y1),(x * y2),(x * y3),(x * y4)*] );
existence
ex b1 being set ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] )
uniqueness
for b1, b2 being set st ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b1 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] ) & ex y1, y2, y3, y4 being Element of REAL st
( y = [*y1,y2,y3,y4*] & b2 = [*(x * y1),(x * y2),(x * y3),(x * y4)*] ) holds
b1 = b2
end;
:: deftheorem Def21 defines * QUATERNI:def 21 :
Lemma185:
for x, y, z, w being Real holds [*x,y,z,w*] = ((x + (y * <i> )) + (z * <j> )) + (w * <k> )
:: deftheorem Def22 defines + QUATERNI:def 22 :
theorem Th36: :: QUATERNI:36
:: deftheorem Def23 defines * QUATERNI:def 23 :
theorem Th37: :: QUATERNI:37
theorem Th38: :: QUATERNI:38
theorem Th39: :: QUATERNI:39
theorem Th40: :: QUATERNI:40
:: deftheorem Def24 defines - QUATERNI:def 24 :
theorem Th41: :: QUATERNI:41
:: deftheorem Def25 defines - QUATERNI:def 25 :
theorem Th42: :: QUATERNI:42
:: deftheorem Def26 defines *' QUATERNI:def 26 :
theorem Th43: :: QUATERNI:43
theorem Th44: :: QUATERNI:44
theorem Th45: :: QUATERNI:45
theorem Th46: :: QUATERNI:46
theorem Th47: :: QUATERNI:47
theorem Th48: :: QUATERNI:48
theorem Th49: :: QUATERNI:49
theorem Th50: :: QUATERNI:50
theorem Th51: :: QUATERNI:51
theorem Th52: :: QUATERNI:52
theorem Th53: :: QUATERNI:53
Lemma190:
for z1, z2 being quaternion number holds
( Rea (z1 + z2) = (Rea z1) + (Rea z2) & Im1 (z1 + z2) = (Im1 z1) + (Im1 z2) & Im2 (z1 + z2) = (Im2 z1) + (Im2 z2) & Im3 (z1 + z2) = (Im3 z1) + (Im3 z2) )
theorem Th54: :: QUATERNI:54
theorem Th55: :: QUATERNI:55
Lemma192:
for z1, z2 being quaternion number holds
( Rea (z1 - z2) = (Rea z1) - (Rea z2) & Im1 (z1 - z2) = (Im1 z1) - (Im1 z2) & Im2 (z1 - z2) = (Im2 z1) - (Im2 z2) & Im3 (z1 - z2) = (Im3 z1) - (Im3 z2) )
theorem Th56: :: QUATERNI:56
theorem Th57: :: QUATERNI:57
theorem Th58: :: QUATERNI:58
theorem Th59: :: QUATERNI:59
theorem Th60: :: QUATERNI:60
theorem Th61: :: QUATERNI:61
theorem Th62: :: QUATERNI:62
theorem Th63: :: QUATERNI:63
theorem Th64: :: QUATERNI:64
:: deftheorem Def27 defines |. QUATERNI:def 27 :
theorem Th65: :: QUATERNI:65
theorem Th66: :: QUATERNI:66
theorem Th67: :: QUATERNI:67
theorem Th68: :: QUATERNI:68
theorem Th69: :: QUATERNI:69
theorem Th70: :: QUATERNI:70
theorem Th71: :: QUATERNI:71
theorem Th72: :: QUATERNI:72
theorem Th73: :: QUATERNI:73
Lemma200:
for a, b, c, d being real number holds (((a ^2 ) + (b ^2 )) + (c ^2 )) + (d ^2 ) >= 0
theorem Th74: :: QUATERNI:74
Lemma201:
for a, b, c, d being Element of REAL holds a ^2 <= (((a ^2 ) + (b ^2 )) + (c ^2 )) + (d ^2 )
Lemma202:
for a, b, c, d being real number holds c ^2 <= (((a ^2 ) + (b ^2 )) + (c ^2 )) + (d ^2 )
Lemma203:
for d, a, b, c being Element of REAL holds d ^2 <= (((a ^2 ) + (b ^2 )) + (c ^2 )) + (d ^2 )
theorem Th75: :: QUATERNI:75
theorem Th76: :: QUATERNI:76
theorem Th77: :: QUATERNI:77
theorem Th78: :: QUATERNI:78
Lemma204:
for z1, z2 being quaternion number holds
( Rea (z1 + z2) = (Rea z1) + (Rea z2) & Im1 (z1 + z2) = (Im1 z1) + (Im1 z2) & Im2 (z1 + z2) = (Im2 z1) + (Im2 z2) & Im3 (z1 + z2) = (Im3 z1) + (Im3 z2) )
Lemma205:
for a, b being Element of REAL st a >= b ^2 holds
sqrt a >= b
Lemma206:
for a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6 being real number st a1 >= b1 & a2 >= b2 & a3 >= b3 & a4 >= b4 & a5 >= b5 & a6 >= b6 holds
((((a1 + a2) + a3) + a4) + a5) + a6 >= ((((b1 + b2) + b3) + b4) + b5) + b6
Lemma219:
for a, b being Element of REAL st a >= 0 & b >= 0 & a ^2 >= b ^2 holds
a >= b
Lemma220:
for m1, m2, m4, m3, n1, n2, n3, n4 being real number holds ((sqrt ((((m1 ^2 ) + (m2 ^2 )) + (m3 ^2 )) + (m4 ^2 ))) + (sqrt ((((n1 ^2 ) + (n2 ^2 )) + (n3 ^2 )) + (n4 ^2 )))) ^2 = ((((((((m1 ^2 ) + (m2 ^2 )) + (m3 ^2 )) + (m4 ^2 )) + (n1 ^2 )) + (n2 ^2 )) + (n3 ^2 )) + (n4 ^2 )) + (2 * (sqrt (((((m1 ^2 ) + (m2 ^2 )) + (m3 ^2 )) + (m4 ^2 )) * ((((n1 ^2 ) + (n2 ^2 )) + (n3 ^2 )) + (n4 ^2 )))))
Lemma221:
for m1, m2, m3, m4, n1, n2, n3, n4 being real number holds (sqrt (((((m1 + n1) ^2 ) + ((m2 + n2) ^2 )) + ((m3 + n3) ^2 )) + ((m4 + n4) ^2 ))) ^2 = ((((m1 + n1) ^2 ) + ((m2 + n2) ^2 )) + ((m3 + n3) ^2 )) + ((m4 + n4) ^2 )
theorem Th79: :: QUATERNI:79
theorem Th80: :: QUATERNI:80
Lemma230:
for z1, z2 being quaternion number holds z1 = (z1 + z2) - z2
Lemma231:
for z1, z2 being quaternion number holds z1 = (z1 - z2) + z2
Lemma232:
for z1, z2 being quaternion number holds z1 - z2 = [*((Rea z1) - (Rea z2)),((Im1 z1) - (Im1 z2)),((Im2 z1) - (Im2 z2)),((Im3 z1) - (Im3 z2))*]
Lemma233:
for z1, z2 being quaternion number holds
( Rea (z1 - z2) = (Rea z1) - (Rea z2) & Im1 (z1 - z2) = (Im1 z1) - (Im1 z2) & Im2 (z1 - z2) = (Im2 z1) - (Im2 z2) & Im3 (z1 - z2) = (Im3 z1) - (Im3 z2) )
Lemma234:
for z1, z2 being quaternion number holds z1 - z2 = - (z2 - z1)
Lemma235:
for z1, z2 being quaternion number st z1 - z2 = 0 holds
z1 = z2
theorem Th81: :: QUATERNI:81
theorem Th82: :: QUATERNI:82
theorem Th83: :: QUATERNI:83
theorem Th84: :: QUATERNI:84
Lemma238:
for z, z1, z2 being quaternion number holds z1 - z2 = (z1 - z) + (z - z2)
theorem Th85: :: QUATERNI:85
theorem Th86: :: QUATERNI:86
theorem Th87: :: QUATERNI:87
theorem Th88: :: QUATERNI:88
theorem Th89: :: QUATERNI:89