:: INT_3 semantic presentation
definition
redefine func multint -> M6(
K19(
INT ,
INT ),
INT )
means :
Def1:
:: INT_3:def 1
for b
1, b
2 being
Element of
INT holds a
1 . b
1,b
2 = multreal . b
1,b
2;
compatibility
for b1 being M6(K19(INT ,INT ), INT ) holds
( b1 = multint iff for b2, b3 being Element of INT holds b1 . b2,b3 = multreal . b2,b3 )
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 :
Lemma2:
for b1 being Element of INT.Ring holds b1 in REAL
set c1 = INT.Ring ;
Lemma3:
0 = 0. INT.Ring
Lemma4:
for b1, b2 being Element of INT.Ring
for b3, b4 being Integer holds
( b3 = b1 & b4 = b2 implies b1 * b2 = b3 * b4 )
Lemma5:
for b1, b2 being Element of INT.Ring holds
( b1 = 1 implies ( b2 * b1 = b2 & b1 * b2 = b2 ) )
Lemma6:
1. INT.Ring = 1
Lemma7:
for b1 being Element of INT.Ring
for b2 being Integer holds
( b2 = b1 implies - b2 = - b1 )
:: deftheorem Def4 defines <= INT_3:def 4 :
Lemma9:
for b1, b2 being Element of INT.Ring holds
( b1 < b2 iff ( b1 <= b2 & b1 <> b2 ) )
:: deftheorem Def5 defines abs INT_3:def 5 :
:: deftheorem Def6 defines absint INT_3:def 6 :
theorem Th1: :: INT_3:1
Lemma13:
for b1 being Integer holds
( b1 = 0 or absreal . b1 >= 1 )
Lemma14:
for b1, b2 being Element of INT.Ring
for b3, b4 being Integer holds
( b3 = b1 & b4 = b2 implies b1 + b2 = b3 + b4 )
Lemma15:
for b1, b2 being Element of INT.Ring holds
( b2 <> 0. INT.Ring implies for b3 being Integer holds
not ( b3 = b2 & 0 <= b3 & ( for b4, b5 being Element of INT.Ring holds
not ( b1 = (b4 * b2) + b5 & not ( not b5 = 0. INT.Ring & not absint . b5 < absint . b2 ) ) ) ) )
Lemma16:
for b1, b2 being Element of INT.Ring holds
( b2 <> 0. INT.Ring implies for b3 being Integer holds
not ( b3 = b2 & 0 <= b3 & ( for b4, b5 being Element of INT.Ring holds
not ( b1 = (b4 * b2) + b5 & 0. INT.Ring <= b5 & b5 < abs b2 ) ) ) )
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 :
Lemma21:
for b1 being non empty associative commutative right_zeroed left_unital Field-like doubleLoopStr
for b2 being Function of the carrier of b1, NAT
for b3, b4 being Element of b1 holds
not ( b4 <> 0. b1 & ( for b5, b6 being Element of b1 holds
not ( b3 = (b5 * b4) + b6 & not ( not b6 = 0. b1 & not b2 . b6 < b2 . b4 ) ) ) )
:: 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 b
1, b
2 being
Integer holds
( ( 0
<= b
2 & b
2 < b
1 implies b
2 mod b
1 = b
2 ) & ( 0
> b
2 & b
2 >= - b
1 implies b
2 mod b
1 = b
1 + b
2 ) )
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 :
Lemma36:
for b1 being natural number holds
( b1 > 0 implies for b2 being natural number holds
( b2 < b1 implies for b3 being natural number holds
( b3 = b1 - b2 implies (b1 - b2) mod b1 = b3 mod b1 ) ) )
theorem Th17: :: INT_3:17
Lemma38:
for b1, b2 being natural number holds
not ( b2 <> 0 & ( for b3 being Nat holds
not ( b3 * b2 <= b1 & b1 < (b3 + 1) * b2 ) ) )
theorem Th18: :: INT_3:18
theorem Th19: :: INT_3:19
definition
let c
2 be
natural number ;
func INT.Ring c
1 -> doubleLoopStr equals :: INT_3:def 13
doubleLoopStr(#
(Segm a1),
(addint a1),
(multint a1),
(In 1,(Segm a1)),
(In 0,(Segm a1)) #);
coherence
doubleLoopStr(# (Segm c2),(addint c2),(multint c2),(In 1,(Segm c2)),(In 0,(Segm c2)) #) is doubleLoopStr
;
end;
:: deftheorem Def13 defines INT.Ring INT_3:def 13 :
theorem Th20: :: INT_3:20
Lemma42:
for b1 being natural number holds
( 0 < b1 implies for b2, b3 being Element of (INT.Ring b1) holds
( b2 = 1 implies ( b3 * b2 = b3 & b2 * b3 = b3 ) ) )
Lemma44:
for b1 being natural number holds
( 1 < b1 implies 1. (INT.Ring b1) = 1 )
theorem Th21: :: INT_3:21
Lemma46:
for b1 being natural number holds
( b1 > 0 implies 0. (INT.Ring b1) = 0 )
theorem Th22: :: INT_3:22
theorem Th23: :: INT_3:23
theorem Th24: :: INT_3:24