:: SYMSP_1 semantic presentation
:: deftheorem Def1 defines _|_ SYMSP_1:def 1 :
set X = {0};
reconsider o = 0 as Element of {0} by TARSKI:def 1;
deffunc H1( Element of {0}, Element of {0}) -> Element of {0} = o;
consider md being BinOp of {0} such that
Lemma20:
for x, y being Element of {0} holds md . x,y = H1(x,y)
from BINOP_1:sch 4();
registration
let F be
Field;
let X be non
empty set ;
let md be
BinOp of
X;
let o be
Element of
X;
let mF be
Function of
[:the carrier of F,X:],
X;
let mo be
Relation of
X;
cluster SymStr(#
a2,
a3,
a4,
a5,
a6 #)
-> non
empty ;
coherence
not SymStr(# X,md,o,mF,mo #) is empty
end;
Lemma29:
for F being Field
for mF being Function of [:the carrier of F,{0}:],{0}
for mo being Relation of {0} holds
( SymStr(# {0},md,o,mF,mo #) is Abelian & SymStr(# {0},md,o,mF,mo #) is add-associative & SymStr(# {0},md,o,mF,mo #) is right_zeroed & SymStr(# {0},md,o,mF,mo #) is right_complementable )
E40:
now
let F be
Field;
let mF be
Function of
[:the carrier of F,{0}:],
{0};
assume E27:
for
a being
Element of
F for
x being
Element of
{0} holds
mF . a,
x = o
;
let mo be
Relation of
{0};
let MPS be non
empty Abelian add-associative right_zeroed right_complementable SymStr of
F;
assume E31:
MPS = SymStr(#
{0},
md,
o,
mF,
mo #)
;
thus
MPS is
VectSp-like
proof
for
x,
y being
Element of
F for
v,
w being
Element of
MPS holds
(
x * (v + w) = (x * v) + (x * w) &
(x + y) * v = (x * v) + (y * v) &
(x * y) * v = x * (y * v) &
(1. F) * v = v )
proof
let x be
Element of
F,
y be
Element of
F;
let v be
Element of
MPS,
w be
Element of
MPS;
E32:
x * (v + w) = (x * v) + (x * w)
proof
E34:
v + w = md . v,
w
by , RLVECT_1:5;
reconsider v =
v,
w =
w as
Element of
{0} by ;
E35:
md . v,
w = o
by ;
reconsider v =
v,
w =
w as
Element of
MPS ;
E36:
x * (v + w) = mF . x,
o
by , , , VECTSP_1:def 24;
E37:
x * (v + w) = o
by , ;
mF . x,
v = o
by ;
then E38:
x * v = o
by , VECTSP_1:def 24;
mF . x,
w = o
by ;
then E39:
x * w = o
by , VECTSP_1:def 24;
o = 0. MPS
by , RLVECT_1:def 2;
hence
x * (v + w) = (x * v) + (x * w)
by , , , RLVECT_1:10;
end;
E44:
(x + y) * v = (x * v) + (y * v)
proof
set z =
x + y;
E45:
(x + y) * v = mF . (x + y),
v
by , VECTSP_1:def 24;
reconsider v =
v as
Element of
MPS ;
reconsider v =
v as
Element of
MPS ;
E46:
(x + y) * v = o
by , , ;
reconsider v =
v as
Element of
MPS ;
E47:
mF . x,
v = o
by , ;
reconsider v =
v as
Element of
MPS ;
E48:
x * v = o
by , , VECTSP_1:def 24;
reconsider v =
v as
Element of
MPS ;
E49:
mF . y,
v = o
by , ;
reconsider v =
v as
Element of
MPS ;
E50:
y * v = o
by , , VECTSP_1:def 24;
o = 0. MPS
by , RLVECT_1:def 2;
hence
(x + y) * v = (x * v) + (y * v)
by , , , RLVECT_1:10;
end;
E51:
(x * y) * v = x * (y * v)
proof
set z =
x * y;
E52:
(x * y) * v = mF . (x * y),
v
by , VECTSP_1:def 24;
reconsider v =
v as
Element of
MPS ;
reconsider v =
v as
Element of
MPS ;
E53:
(x * y) * v = o
by , , ;
reconsider v =
v as
Element of
MPS ;
E54:
mF . y,
v = o
by , ;
reconsider v =
v as
Element of
MPS ;
y * v = o
by , , VECTSP_1:def 24;
then E55:
x * (y * v) = mF . x,
o
by , VECTSP_1:def 24;
thus
(x * y) * v = x * (y * v)
by , , ;
end;
(1. F) * v = v
hence
(
x * (v + w) = (x * v) + (x * w) &
(x + y) * v = (x * v) + (y * v) &
(x * y) * v = x * (y * v) &
(1. F) * v = v )
by , , ;
end;
hence
MPS is
VectSp-like
by VECTSP_1:def 26;
end;
end;
E58:
now
let F be
Field;
let mF be
Function of
[:the carrier of F,{0}:],
{0};
assume
for
a being
Element of
F for
x being
Element of
{0} holds
mF . a,
x = o
;
set CV =
[:{0},{0}:];
let mo be
Relation of
{0};
assume E27:
for
x being
set holds
(
x in mo iff (
x in [:{0},{0}:] & ex
a,
b being
Element of
{0} st
(
x = [a,b] &
b = o ) ) )
;
let MPS be non
empty Abelian add-associative right_zeroed right_complementable SymStr of
F;
assume E31:
MPS = SymStr(#
{0},
md,
o,
mF,
mo #)
;
E32:
for
a,
b being
Element of
MPS holds
(
a _|_ b iff (
[a,b] in [:{0},{0}:] & ex
c,
d being
Element of
{0} st
(
[a,b] = [c,d] &
d = o ) ) )
E34:
for
a,
b being
Element of
MPS holds
(
a _|_ b iff
b = o )
proof
let a be
Element of
MPS,
b be
Element of
MPS;
E35:
(
a _|_ b implies
b = o )
(
b = o implies
a _|_ b )
proof
assume E36:
b = o
;
consider c being
Element of
MPS,
d being
Element of
MPS such that E37:
(
c = a &
d = b )
;
[a,b] = [c,d]
by ;
hence
a _|_ b
by , , ;
end;
hence
(
a _|_ b iff
b = o )
by ;
end;
0. MPS = o
by , TARSKI:def 1;
hence
for
a being
Element of
MPS st
a <> 0. MPS holds
ex
p being
Element of
MPS st not
p _|_ a
by , TARSKI:def 1;
thus
for
a,
b being
Element of
MPS for
l being
Element of
F st
a _|_ b holds
l * a _|_ b
thus
for
a,
b,
c being
Element of
MPS st
b _|_ a &
c _|_ a holds
b + c _|_ a
thus
for
a,
b,
x being
Element of
MPS st not
b _|_ a holds
ex
k being
Element of
F st
x - (k * b) _|_ a
let a be
Element of
MPS,
b be
Element of
MPS,
c be
Element of
MPS;
assume
(
a _|_ b + c &
b _|_ c + a )
;
assume
not
c _|_ a + b
;
then
a + b <> o
by ;
hence
contradiction
by , TARSKI:def 1;
end;
:: deftheorem Def2 defines SymSp-like SYMSP_1:def 2 :
theorem Th1: :: SYMSP_1:1
canceled;
theorem Th2: :: SYMSP_1:2
canceled;
theorem Th3: :: SYMSP_1:3
canceled;
theorem Th4: :: SYMSP_1:4
canceled;
theorem Th5: :: SYMSP_1:5
canceled;
theorem Th6: :: SYMSP_1:6
canceled;
theorem Th7: :: SYMSP_1:7
canceled;
theorem Th8: :: SYMSP_1:8
canceled;
theorem Th9: :: SYMSP_1:9
canceled;
theorem Th10: :: SYMSP_1:10
canceled;
theorem Th11: :: SYMSP_1:11
theorem Th12: :: SYMSP_1:12
theorem Th13: :: SYMSP_1:13
theorem Th14: :: SYMSP_1:14
theorem Th15: :: SYMSP_1:15
theorem Th16: :: SYMSP_1:16
theorem Th17: :: SYMSP_1:17
canceled;
theorem Th18: :: SYMSP_1:18
canceled;
theorem Th19: :: SYMSP_1:19
theorem Th20: :: SYMSP_1:20
theorem Th21: :: SYMSP_1:21
theorem Th22: :: SYMSP_1:22
theorem Th23: :: SYMSP_1:23
theorem Th24: :: SYMSP_1:24
theorem Th25: :: SYMSP_1:25
:: deftheorem Def3 SYMSP_1:def 3 :
canceled;
:: deftheorem Def4 SYMSP_1:def 4 :
canceled;
:: deftheorem Def5 SYMSP_1:def 5 :
canceled;
:: deftheorem Def6 defines ProJ SYMSP_1:def 6 :
theorem Th26: :: SYMSP_1:26
canceled;
theorem Th27: :: SYMSP_1:27
theorem Th28: :: SYMSP_1:28
theorem Th29: :: SYMSP_1:29
theorem Th30: :: SYMSP_1:30
theorem Th31: :: SYMSP_1:31
theorem Th32: :: SYMSP_1:32
for
F being
Field for
S being
SymSp of
F for
b,
a,
p,
c being
Element of
S st not
b _|_ a &
p _|_ a holds
(
ProJ a,
(b + p),
c = ProJ a,
b,
c &
ProJ a,
b,
(c + p) = ProJ a,
b,
c )
theorem Th33: :: SYMSP_1:33
theorem Th34: :: SYMSP_1:34
theorem Th35: :: SYMSP_1:35
theorem Th36: :: SYMSP_1:36
theorem Th37: :: SYMSP_1:37
theorem Th38: :: SYMSP_1:38
theorem Th39: :: SYMSP_1:39
theorem Th40: :: SYMSP_1:40
theorem Th41: :: SYMSP_1:41
for
F being
Field for
S being
SymSp of
F for
a,
p,
q,
b being
Element of
S st
(1. F) + (1. F) <> 0. F & not
a _|_ p & not
a _|_ q & not
b _|_ p & not
b _|_ q holds
(ProJ a,p,q) * (ProJ b,q,p) = (ProJ p,a,b) * (ProJ q,b,a)
theorem Th42: :: SYMSP_1:42
for
F being
Field for
S being
SymSp of
F for
p,
a,
x,
q being
Element of
S st
(1. F) + (1. F) <> 0. F & not
p _|_ a & not
p _|_ x & not
q _|_ a & not
q _|_ x holds
(ProJ a,q,p) * (ProJ p,a,x) = (ProJ x,q,p) * (ProJ q,a,x)
theorem Th43: :: SYMSP_1:43
for
F being
Field for
S being
SymSp of
F for
p,
a,
x,
q,
b,
y being
Element of
S st
(1. F) + (1. F) <> 0. F & not
p _|_ a & not
p _|_ x & not
q _|_ a & not
q _|_ x & not
b _|_ a holds
((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,y) = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y)
theorem Th44: :: SYMSP_1:44
for
F being
Field for
S being
SymSp of
F for
a,
p,
x,
y being
Element of
S st not
a _|_ p & not
x _|_ p & not
y _|_ p holds
(ProJ p,a,x) * (ProJ x,p,y) = (- (ProJ p,a,y)) * (ProJ y,p,x)
definition
let F be
Field;
let S be
SymSp of
F;
let x be
Element of
S;
let y be
Element of
S;
let a be
Element of
S;
let b be
Element of
S;
assume that E27:
not
b _|_ a
and E31:
(1. F) + (1. F) <> 0. F
;
func PProJ c5,
c6,
c3,
c4 -> Element of
a1 means :
Def7:
:: SYMSP_1:def 7
for
q being
Element of
o st not
q _|_ S & not
q _|_ md holds
it = ((ProJ S,x,q) * (ProJ q,S,md)) * (ProJ md,q,F) if ex
p being
Element of
o st
( not
p _|_ S & not
p _|_ md )
it = 0. X if for
p being
Element of
o holds
(
p _|_ S or
p _|_ md )
;
existence
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) implies ex b1 being Element of F st
for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) implies ex b1 being Element of F st b1 = 0. F ) )
uniqueness
for b1, b2 being Element of F holds
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b2 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) implies b1 = b2 ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) & b1 = 0. F & b2 = 0. F implies b1 = b2 ) )
consistency
for b1 being Element of F st ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) holds
( ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) iff b1 = 0. F )
;
end;
:: deftheorem Def7 defines PProJ SYMSP_1:def 7 :
for
F being
Field for
S being
SymSp of
F for
x,
y,
a,
b being
Element of
S st not
b _|_ a &
(1. F) + (1. F) <> 0. F holds
for
b7 being
Element of
F holds
( ( ex
p being
Element of
S st
( not
p _|_ a & not
p _|_ x ) implies (
b7 = PProJ a,
b,
x,
y iff for
q being
Element of
S st not
q _|_ a & not
q _|_ x holds
b7 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) ) & ( ( for
p being
Element of
S holds
(
p _|_ a or
p _|_ x ) ) implies (
b7 = PProJ a,
b,
x,
y iff
b7 = 0. F ) ) );
theorem Th45: :: SYMSP_1:45
canceled;
theorem Th46: :: SYMSP_1:46
canceled;
theorem Th47: :: SYMSP_1:47
Lemma111:
for F being Field
for S being SymSp of F
for x being Element of S holds x _|_ 0. S
theorem Th48: :: SYMSP_1:48
theorem Th49: :: SYMSP_1:49
theorem Th50: :: SYMSP_1:50
theorem Th51: :: SYMSP_1:51
for
F being
Field for
S being
SymSp of
F for
b,
a,
x,
y,
z being
Element of
S st
(1. F) + (1. F) <> 0. F & not
b _|_ a holds
PProJ a,
b,
x,
(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)