:: INT_3 semantic presentation
definition
redefine func multint -> M6(
K19(
INT ,
INT ),
INT )
means :
Def1:
:: INT_3:def 1
for
a,
b being
Element of
INT holds
it . a,
b = multreal . a,
b;
compatibility
for b1 being M6(K19(INT ,INT ), INT ) holds
( b1 = multint iff for a, b being Element of INT holds b1 . a,b = multreal . a,b )
end;
:: deftheorem Def1 defines multint INT_3:def 1 :
:: deftheorem Def2 defines compint INT_3:def 2 :
definition
func INT.Ring -> doubleLoopStr equals :: INT_3:def 3
doubleLoopStr(#
INT ,
addint ,
multint ,
(In 1,INT ),
(In 0,INT ) #);
coherence
doubleLoopStr(# INT ,addint ,multint ,(In 1,INT ),(In 0,INT ) #) is doubleLoopStr
;
end;
:: deftheorem Def3 defines INT.Ring INT_3:def 3 :
Lemma26:
for x being Element of INT.Ring holds x in REAL
set M = INT.Ring ;
Lemma29:
0 = 0. INT.Ring
Lemma30:
for a, b being Element of INT.Ring
for a', b' being Integer st a' = a & b' = b holds
a * b = a' * b'
Lemma33:
for e, h being Element of INT.Ring st e = 1 holds
( h * e = h & e * h = h )
Lemma36:
1. INT.Ring = 1
Lemma61:
for a being Element of INT.Ring
for a' being Integer st a' = a holds
- a' = - a
:: deftheorem Def4 defines <= INT_3:def 4 :
Lemma64:
for a, b being Element of INT.Ring holds
( a < b iff ( a <= b & a <> b ) )
:: deftheorem Def5 defines abs INT_3:def 5 :
:: deftheorem Def6 defines absint INT_3:def 6 :
theorem Th1: :: INT_3:1
Lemma74:
for a being Integer holds
( a = 0 or absreal . a >= 1 )
Lemma75:
for a, b being Element of INT.Ring
for a', b' being Integer st a' = a & b' = b holds
a + b = a' + b'
Lemma76:
for a, b being Element of INT.Ring st b <> 0. INT.Ring holds
for b' being Integer st b' = b & 0 <= b' holds
ex q, r being Element of INT.Ring st
( a = (q * b) + r & ( r = 0. INT.Ring or absint . r < absint . b ) )
Lemma87:
for a, b being Element of INT.Ring st b <> 0. INT.Ring holds
for b' being Integer st b' = b & 0 <= b' holds
ex q, r being Element of INT.Ring st
( a = (q * b) + r & 0. INT.Ring <= r & r < abs b )
theorem Th2: :: INT_3:2
:: deftheorem Def7 defines div INT_3:def 7 :
:: deftheorem Def8 defines mod INT_3:def 8 :
theorem Th3: :: INT_3:3
:: deftheorem Def9 defines Euclidian INT_3:def 9 :
Lemma116:
for F being non empty associative commutative right_zeroed left_unital Field-like doubleLoopStr
for f being Function of the carrier of F, NAT
for a, b being Element of F st b <> 0. F holds
ex q, r being Element of F st
( a = (q * b) + r & ( r = 0. F or f . r < f . b ) )
:: deftheorem Def10 defines DegreeFunction INT_3:def 10 :
theorem Th4: :: INT_3:4
theorem Th5: :: INT_3:5
theorem Th6: :: INT_3:6
theorem Th7: :: INT_3:7
canceled;
theorem Th8: :: INT_3:8
theorem Th9: :: INT_3:9
theorem Th10: :: INT_3:10
for
n,
a being
Integer holds
( ( 0
<= a &
a < n implies
a mod n = a ) & ( 0
> a &
a >= - n implies
a mod n = n + a ) )
theorem Th11: :: INT_3:11
theorem Th12: :: INT_3:12
theorem Th13: :: INT_3:13
theorem Th14: :: INT_3:14
theorem Th15: :: INT_3:15
theorem Th16: :: INT_3:16
:: deftheorem Def11 defines multint INT_3:def 11 :
:: deftheorem Def12 defines compint INT_3:def 12 :
Lemma157:
for n being natural number st n > 0 holds
for a being natural number st a < n holds
for b being natural number st b = n - a holds
(n - a) mod n = b mod n
theorem Th17: :: INT_3:17
Lemma159:
for a, b being natural number st b <> 0 holds
ex k being Element of NAT st
( k * b <= a & a < (k + 1) * b )
theorem Th18: :: INT_3:18
theorem Th19: :: INT_3:19
definition
let n be
natural number ;
func INT.Ring c1 -> doubleLoopStr equals :: INT_3:def 13
doubleLoopStr(#
(Segm M),
(addint M),
(multint M),
(In 1,(Segm M)),
(In 0,(Segm M)) #);
coherence
doubleLoopStr(# (Segm n),(addint n),(multint n),(In 1,(Segm n)),(In 0,(Segm n)) #) is doubleLoopStr
;
end;
:: deftheorem Def13 defines INT.Ring INT_3:def 13 :
theorem Th20: :: INT_3:20
Lemma164:
for n being natural number st 0 < n holds
for e, h being Element of (INT.Ring n) st e = 1 holds
( h * e = h & e * h = h )
Lemma167:
for n being natural number st 1 < n holds
1. (INT.Ring n) = 1
theorem Th21: :: INT_3:21
Lemma178:
for p being natural number st p > 0 holds
0. (INT.Ring p) = 0
theorem Th22: :: INT_3:22
theorem Th23: :: INT_3:23
theorem Th24: :: INT_3:24