:: GROUP_7 semantic presentation
theorem Th1: :: GROUP_7:1
theorem Th2: :: GROUP_7:2
for b
1, b
2, b
3, b
4 being
set holds
(
<*b1,b2*> = <*b3,b4*> implies ( b
1 = b
3 & b
2 = b
4 ) )
theorem Th3: :: GROUP_7:3
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
<*b1,b2,b3*> = <*b4,b5,b6*> implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
:: deftheorem Def1 defines HGrStr-yielding GROUP_7:def 1 :
definition
let c
1 be
set ;
let c
2 be
HGrStr-Family of c
1;
func product c
2 -> strict HGrStr means :
Def2:
:: GROUP_7:def 2
( the
carrier of a
3 = product (Carrier a2) & ( for b
1, b
2 being
Element of
product (Carrier a2)for b
3 being
set holds
not ( b
3 in a
1 & ( for b
4 being non
empty HGrStr for b
5 being
Function holds
not ( b
4 = a
2 . b
3 & b
5 = the
mult of a
3 . b
1,b
2 & b
5 . b
3 = the
mult of b
4 . (b1 . b3),
(b2 . b3) ) ) ) ) );
existence
ex b1 being strict HGrStr st
( the carrier of b1 = product (Carrier c2) & ( for b2, b3 being Element of product (Carrier c2)
for b4 being set holds
not ( b4 in c1 & ( for b5 being non empty HGrStr
for b6 being Function holds
not ( b5 = c2 . b4 & b6 = the mult of b1 . b2,b3 & b6 . b4 = the mult of b5 . (b2 . b4),(b3 . b4) ) ) ) ) )
uniqueness
for b1, b2 being strict HGrStr holds
( the carrier of b1 = product (Carrier c2) & ( for b3, b4 being Element of product (Carrier c2)
for b5 being set holds
not ( b5 in c1 & ( for b6 being non empty HGrStr
for b7 being Function holds
not ( b6 = c2 . b5 & b7 = the mult of b1 . b3,b4 & b7 . b5 = the mult of b6 . (b3 . b5),(b4 . b5) ) ) ) ) & the carrier of b2 = product (Carrier c2) & ( for b3, b4 being Element of product (Carrier c2)
for b5 being set holds
not ( b5 in c1 & ( for b6 being non empty HGrStr
for b7 being Function holds
not ( b6 = c2 . b5 & b7 = the mult of b2 . b3,b4 & b7 . b5 = the mult of b6 . (b3 . b5),(b4 . b5) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines product GROUP_7:def 2 :
theorem Th4: :: GROUP_7:4
:: deftheorem Def3 defines Group-like GROUP_7:def 3 :
:: deftheorem Def4 defines associative GROUP_7:def 4 :
:: deftheorem Def5 defines commutative GROUP_7:def 5 :
:: deftheorem Def6 defines Group-like GROUP_7:def 6 :
:: deftheorem Def7 defines associative GROUP_7:def 7 :
:: deftheorem Def8 defines commutative GROUP_7:def 8 :
theorem Th5: :: GROUP_7:5
theorem Th6: :: GROUP_7:6
theorem Th7: :: GROUP_7:7
theorem Th8: :: GROUP_7:8
theorem Th9: :: GROUP_7:9
theorem Th10: :: GROUP_7:10
theorem Th11: :: GROUP_7:11
:: deftheorem Def9 defines sum GROUP_7:def 9 :
theorem Th12: :: GROUP_7:12
theorem Th13: :: GROUP_7:13
theorem Th14: :: GROUP_7:14
theorem Th15: :: GROUP_7:15
theorem Th16: :: GROUP_7:16
theorem Th17: :: GROUP_7:17
theorem Th18: :: GROUP_7:18
theorem Th19: :: GROUP_7:19
theorem Th20: :: GROUP_7:20
theorem Th21: :: GROUP_7:21
theorem Th22: :: GROUP_7:22
theorem Th23: :: GROUP_7:23
theorem Th24: :: GROUP_7:24
theorem Th25: :: GROUP_7:25
definition
let c
1, c
2, c
3 be non
empty HGrStr ;
redefine func <* as
<*c1,c2,c3*> -> HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is HGrStr-Family of {1,2,3}
by Th25;
end;
theorem Th26: :: GROUP_7:26
definition
let c
1, c
2, c
3 be non
empty Group-like HGrStr ;
redefine func <* as
<*c1,c2,c3*> -> Group-like HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is Group-like HGrStr-Family of {1,2,3}
by Th26;
end;
theorem Th27: :: GROUP_7:27
definition
let c
1, c
2, c
3 be non
empty associative HGrStr ;
redefine func <* as
<*c1,c2,c3*> -> associative HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is associative HGrStr-Family of {1,2,3}
by Th27;
end;
theorem Th28: :: GROUP_7:28
definition
let c
1, c
2, c
3 be non
empty commutative HGrStr ;
redefine func <* as
<*c1,c2,c3*> -> commutative HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is commutative HGrStr-Family of {1,2,3}
by Th28;
end;
theorem Th29: :: GROUP_7:29
definition
let c
1, c
2, c
3 be
Group;
redefine func <* as
<*c1,c2,c3*> -> Group-like associative HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is Group-like associative HGrStr-Family of {1,2,3}
by Th29;
end;
theorem Th30: :: GROUP_7:30
definition
let c
1, c
2, c
3 be
commutative Group;
redefine func <* as
<*c1,c2,c3*> -> Group-like associative commutative HGrStr-Family of
{1,2,3};
coherence
<*c1,c2,c3*> is Group-like associative commutative HGrStr-Family of {1,2,3}
by Th30;
end;
definition
let c
1, c
2, c
3 be non
empty HGrStr ;
let c
4 be
Element of c
1;
let c
5 be
Element of c
2;
let c
6 be
Element of c
3;
redefine func <* as
<*c4,c5,c6*> -> Element of
(product <*a1,a2,a3*>);
coherence
<*c4,c5,c6*> is Element of (product <*c1,c2,c3*>)
end;
theorem Th31: :: GROUP_7:31
theorem Th32: :: GROUP_7:32
theorem Th33: :: GROUP_7:33
for b
1, b
2, b
3 being non
empty HGrStr for b
4, b
5 being
Element of b
1for b
6, b
7 being
Element of b
2for b
8, b
9 being
Element of b
3 holds
<*b4,b6,b8*> * <*b5,b7,b9*> = <*(b4 * b5),(b6 * b7),(b8 * b9)*>
theorem Th34: :: GROUP_7:34
theorem Th35: :: GROUP_7:35
theorem Th36: :: GROUP_7:36
theorem Th37: :: GROUP_7:37
theorem Th38: :: GROUP_7:38
theorem Th39: :: GROUP_7:39
theorem Th40: :: GROUP_7:40
theorem Th41: :: GROUP_7:41
theorem Th42: :: GROUP_7:42