:: GRCAT_1 semantic presentation
theorem Th1: :: GRCAT_1:1
canceled;
theorem Th2: :: GRCAT_1:2
for b
1, b
2, b
3, b
4 being
set holds
not ( b
4 in b
3 & b
3 c= [:b1,b2:] & ( for b
5 being
Element of b
1for b
6 being
Element of b
2 holds
not b
4 = [b5,b6] ) )
theorem Th3: :: GRCAT_1:3
theorem Th4: :: GRCAT_1:4
for b
1 being
Universefor b
2, b
3 being
set holds
( b
2 in b
3 & b
3 in b
1 implies b
2 in b
1 )
scheme :: GRCAT_1:sch 1
s1{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4(
set ,
set )
-> set , P
1[
set ,
set ] } :
ex b
1 being
PartFunc of
[:F1(),F2():],F
3() st
( ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 iff ( b
2 in F
1() & b
3 in F
2() & P
1[b
2,b
3] ) ) ) & ( for b
2, b
3 being
set holds
(
[b2,b3] in dom b
1 implies b
1 . [b2,b3] = F
4(b
2,b
3) ) ) )
provided
E4:
for b
1, b
2 being
set holds
( b
1 in F
1() & b
2 in F
2() & P
1[b
1,b
2] implies F
4(b
1,b
2)
in F
3() )
theorem Th5: :: GRCAT_1:5
theorem Th6: :: GRCAT_1:6
theorem Th7: :: GRCAT_1:7
theorem Th8: :: GRCAT_1:8
:: deftheorem Def1 GRCAT_1:def 1 :
canceled;
:: deftheorem Def2 GRCAT_1:def 2 :
canceled;
:: deftheorem Def3 GRCAT_1:def 3 :
canceled;
:: deftheorem Def4 GRCAT_1:def 4 :
canceled;
:: deftheorem Def5 defines Morphs GRCAT_1:def 5 :
definition
let c
1 be
Category;
let c
2 be non
empty Subset of the
Objects of c
1;
func dom c
2 -> Function of
Morphs a
2,a
2 equals :: GRCAT_1:def 6
the
Dom of a
1 | (Morphs a2);
coherence
the Dom of c1 | (Morphs c2) is Function of Morphs c2,c2
by CAT_2:29;
func cod c
2 -> Function of
Morphs a
2,a
2 equals :: GRCAT_1:def 7
the
Cod of a
1 | (Morphs a2);
coherence
the Cod of c1 | (Morphs c2) is Function of Morphs c2,c2
by CAT_2:29;
func comp c
2 -> PartFunc of
[:(Morphs a2),(Morphs a2):],
Morphs a
2 equals :: GRCAT_1:def 8
the
Comp of a
1 || (Morphs a2);
coherence
the Comp of c1 || (Morphs c2) is PartFunc of [:(Morphs c2),(Morphs c2):], Morphs c2
by CAT_2:29;
func ID c
2 -> Function of a
2,
Morphs a
2 equals :: GRCAT_1:def 9
the
Id of a
1 | a
2;
coherence
the Id of c1 | c2 is Function of c2, Morphs c2
by CAT_2:29;
end;
:: deftheorem Def6 defines dom GRCAT_1:def 6 :
:: deftheorem Def7 defines cod GRCAT_1:def 7 :
:: deftheorem Def8 defines comp GRCAT_1:def 8 :
:: deftheorem Def9 defines ID GRCAT_1:def 9 :
theorem Th9: :: GRCAT_1:9
definition
let c
1 be
Category;
let c
2 be non
empty Subset of the
Objects of c
1;
func cat c
2 -> Subcategory of a
1 equals :: GRCAT_1:def 10
CatStr(# a
2,
(Morphs a2),
(dom a2),
(cod a2),
(comp a2),
(ID a2) #);
coherence
CatStr(# c2,(Morphs c2),(dom c2),(cod c2),(comp c2),(ID c2) #) is Subcategory of c1
end;
:: deftheorem Def10 defines cat GRCAT_1:def 10 :
theorem Th10: :: GRCAT_1:10
:: deftheorem Def11 defines id GRCAT_1:def 11 :
theorem Th11: :: GRCAT_1:11
theorem Th12: :: GRCAT_1:12
:: deftheorem Def12 defines ZeroMap GRCAT_1:def 12 :
:: deftheorem Def13 defines additive GRCAT_1:def 13 :
theorem Th13: :: GRCAT_1:13
theorem Th14: :: GRCAT_1:14
theorem Th15: :: GRCAT_1:15
theorem Th16: :: GRCAT_1:16
:: deftheorem Def14 defines dom GRCAT_1:def 14 :
:: deftheorem Def15 defines cod GRCAT_1:def 15 :
:: deftheorem Def16 defines fun GRCAT_1:def 16 :
theorem Th17: :: GRCAT_1:17
:: deftheorem Def17 defines ZERO GRCAT_1:def 17 :
:: deftheorem Def18 defines GroupMorphism-like GRCAT_1:def 18 :
theorem Th18: :: GRCAT_1:18
:: deftheorem Def19 defines Morphism GRCAT_1:def 19 :
theorem Th19: :: GRCAT_1:19
theorem Th20: :: GRCAT_1:20
theorem Th21: :: GRCAT_1:21
:: deftheorem Def20 defines ID GRCAT_1:def 20 :
theorem Th22: :: GRCAT_1:22
theorem Th23: :: GRCAT_1:23
theorem Th24: :: GRCAT_1:24
theorem Th25: :: GRCAT_1:25
theorem Th26: :: GRCAT_1:26
definition
let c
1, c
2 be
GroupMorphism;
assume E25:
dom c
1 = cod c
2
;
func c
1 * c
2 -> strict GroupMorphism means :
Def21:
:: GRCAT_1:def 21
for b
1, b
2, b
3 being
AddGroupfor b
4 being
Function of b
2,b
3for b
5 being
Function of b
1,b
2 holds
(
GroupMorphismStr(# the
Dom of a
1,the
Cod of a
1,the
Fun of a
1 #)
= GroupMorphismStr(# b
2,b
3,b
4 #) &
GroupMorphismStr(# the
Dom of a
2,the
Cod of a
2,the
Fun of a
2 #)
= GroupMorphismStr(# b
1,b
2,b
5 #) implies a
3 = GroupMorphismStr(# b
1,b
3,
(b4 * b5) #) );
existence
ex b1 being strict GroupMorphism st
for b2, b3, b4 being AddGroup
for b5 being Function of b3,b4
for b6 being Function of b2,b3 holds
( GroupMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = GroupMorphismStr(# b3,b4,b5 #) & GroupMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = GroupMorphismStr(# b2,b3,b6 #) implies b1 = GroupMorphismStr(# b2,b4,(b5 * b6) #) )
uniqueness
for b1, b2 being strict GroupMorphism holds
( ( for b3, b4, b5 being AddGroup
for b6 being Function of b4,b5
for b7 being Function of b3,b4 holds
( GroupMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = GroupMorphismStr(# b4,b5,b6 #) & GroupMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = GroupMorphismStr(# b3,b4,b7 #) implies b1 = GroupMorphismStr(# b3,b5,(b6 * b7) #) ) ) & ( for b3, b4, b5 being AddGroup
for b6 being Function of b4,b5
for b7 being Function of b3,b4 holds
( GroupMorphismStr(# the Dom of c1,the Cod of c1,the Fun of c1 #) = GroupMorphismStr(# b4,b5,b6 #) & GroupMorphismStr(# the Dom of c2,the Cod of c2,the Fun of c2 #) = GroupMorphismStr(# b3,b4,b7 #) implies b2 = GroupMorphismStr(# b3,b5,(b6 * b7) #) ) ) implies b1 = b2 )
end;
:: deftheorem Def21 defines * GRCAT_1:def 21 :
for b
1, b
2 being
GroupMorphism holds
(
dom b
1 = cod b
2 implies for b
3 being
strict GroupMorphism holds
( b
3 = b
1 * b
2 iff for b
4, b
5, b
6 being
AddGroupfor b
7 being
Function of b
5,b
6for b
8 being
Function of b
4,b
5 holds
(
GroupMorphismStr(# the
Dom of b
1,the
Cod of b
1,the
Fun of b
1 #)
= GroupMorphismStr(# b
5,b
6,b
7 #) &
GroupMorphismStr(# the
Dom of b
2,the
Cod of b
2,the
Fun of b
2 #)
= GroupMorphismStr(# b
4,b
5,b
8 #) implies b
3 = GroupMorphismStr(# b
4,b
6,
(b7 * b8) #) ) ) );
theorem Th27: :: GRCAT_1:27
canceled;
theorem Th28: :: GRCAT_1:28
theorem Th29: :: GRCAT_1:29
for b
1, b
2, b
3 being
AddGroupfor b
4 being
Morphism of b
2,b
3for b
5 being
Morphism of b
1,b
2for b
6 being
Function of b
2,b
3for b
7 being
Function of b
1,b
2 holds
( b
4 = GroupMorphismStr(# b
2,b
3,b
6 #) & b
5 = GroupMorphismStr(# b
1,b
2,b
7 #) implies b
4 * b
5 = GroupMorphismStr(# b
1,b
3,
(b6 * b7) #) )
theorem Th30: :: GRCAT_1:30
for b
1, b
2 being
strict GroupMorphism holds
not (
dom b
2 = cod b
1 & ( for b
3, b
4, b
5 being
AddGroupfor b
6 being
Function of b
3,b
4for b
7 being
Function of b
4,b
5 holds
not ( b
1 = GroupMorphismStr(# b
3,b
4,b
6 #) & b
2 = GroupMorphismStr(# b
4,b
5,b
7 #) & b
2 * b
1 = GroupMorphismStr(# b
3,b
5,
(b7 * b6) #) ) ) )
theorem Th31: :: GRCAT_1:31
theorem Th32: :: GRCAT_1:32
theorem Th33: :: GRCAT_1:33
theorem Th34: :: GRCAT_1:34
:: deftheorem Def22 defines Group_DOMAIN-like GRCAT_1:def 22 :
:: deftheorem Def23 defines GroupMorphism_DOMAIN-like GRCAT_1:def 23 :
theorem Th35: :: GRCAT_1:35
canceled;
theorem Th36: :: GRCAT_1:36
canceled;
theorem Th37: :: GRCAT_1:37
:: deftheorem Def24 defines GroupMorphism_DOMAIN GRCAT_1:def 24 :
theorem Th38: :: GRCAT_1:38
theorem Th39: :: GRCAT_1:39
:: deftheorem Def25 defines MapsSet GRCAT_1:def 25 :
:: deftheorem Def26 defines Maps GRCAT_1:def 26 :
definition
let c
1, c
2 be
AddGroup;
func Morphs c
1,c
2 -> GroupMorphism_DOMAIN of a
1,a
2 means :
Def27:
:: GRCAT_1:def 27
for b
1 being
set holds
( b
1 in a
3 iff b
1 is
strict Morphism of a
1,a
2 );
existence
ex b1 being GroupMorphism_DOMAIN of c1,c2 st
for b2 being set holds
( b2 in b1 iff b2 is strict Morphism of c1,c2 )
uniqueness
for b1, b2 being GroupMorphism_DOMAIN of c1,c2 holds
( ( for b3 being set holds
( b3 in b1 iff b3 is strict Morphism of c1,c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is strict Morphism of c1,c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def27 defines Morphs GRCAT_1:def 27 :
:: deftheorem Def28 defines GO GRCAT_1:def 28 :
theorem Th40: :: GRCAT_1:40
for b
1, b
2, b
3 being
set holds
(
GO b
1,b
2 &
GO b
1,b
3 implies b
2 = b
3 )
theorem Th41: :: GRCAT_1:41
:: deftheorem Def29 defines GroupObjects GRCAT_1:def 29 :
theorem Th42: :: GRCAT_1:42
theorem Th43: :: GRCAT_1:43
:: deftheorem Def30 defines Morphs GRCAT_1:def 30 :
:: deftheorem Def31 defines ID GRCAT_1:def 31 :
definition
let c
1 be
Group_DOMAIN;
func dom c
1 -> Function of
Morphs a
1,a
1 means :
Def32:
:: GRCAT_1:def 32
for b
1 being
Element of
Morphs a
1 holds a
2 . b
1 = dom b
1;
existence
ex b1 being Function of Morphs c1,c1 st
for b2 being Element of Morphs c1 holds b1 . b2 = dom b2
uniqueness
for b1, b2 being Function of Morphs c1,c1 holds
( ( for b3 being Element of Morphs c1 holds b1 . b3 = dom b3 ) & ( for b3 being Element of Morphs c1 holds b2 . b3 = dom b3 ) implies b1 = b2 )
func cod c
1 -> Function of
Morphs a
1,a
1 means :
Def33:
:: GRCAT_1:def 33
for b
1 being
Element of
Morphs a
1 holds a
2 . b
1 = cod b
1;
existence
ex b1 being Function of Morphs c1,c1 st
for b2 being Element of Morphs c1 holds b1 . b2 = cod b2
uniqueness
for b1, b2 being Function of Morphs c1,c1 holds
( ( for b3 being Element of Morphs c1 holds b1 . b3 = cod b3 ) & ( for b3 being Element of Morphs c1 holds b2 . b3 = cod b3 ) implies b1 = b2 )
func ID c
1 -> Function of a
1,
Morphs a
1 means :
Def34:
:: GRCAT_1:def 34
for b
1 being
Element of a
1 holds a
2 . b
1 = ID b
1;
existence
ex b1 being Function of c1, Morphs c1 st
for b2 being Element of c1 holds b1 . b2 = ID b2
uniqueness
for b1, b2 being Function of c1, Morphs c1 holds
( ( for b3 being Element of c1 holds b1 . b3 = ID b3 ) & ( for b3 being Element of c1 holds b2 . b3 = ID b3 ) implies b1 = b2 )
end;
:: deftheorem Def32 defines dom GRCAT_1:def 32 :
:: deftheorem Def33 defines cod GRCAT_1:def 33 :
:: deftheorem Def34 defines ID GRCAT_1:def 34 :
theorem Th44: :: GRCAT_1:44
theorem Th45: :: GRCAT_1:45
definition
let c
1 be
Group_DOMAIN;
func comp c
1 -> PartFunc of
[:(Morphs a1),(Morphs a1):],
Morphs a
1 means :
Def35:
:: GRCAT_1:def 35
( ( for b
1, b
2 being
Element of
Morphs a
1 holds
(
[b1,b2] in dom a
2 iff
dom b
1 = cod b
2 ) ) & ( for b
1, b
2 being
Element of
Morphs a
1 holds
(
[b1,b2] in dom a
2 implies a
2 . [b1,b2] = b
1 * b
2 ) ) );
existence
ex b1 being PartFunc of [:(Morphs c1),(Morphs c1):], Morphs c1 st
( ( for b2, b3 being Element of Morphs c1 holds
( [b2,b3] in dom b1 iff dom b2 = cod b3 ) ) & ( for b2, b3 being Element of Morphs c1 holds
( [b2,b3] in dom b1 implies b1 . [b2,b3] = b2 * b3 ) ) )
uniqueness
for b1, b2 being PartFunc of [:(Morphs c1),(Morphs c1):], Morphs c1 holds
( ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b1 iff dom b3 = cod b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b1 implies b1 . [b3,b4] = b3 * b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b2 iff dom b3 = cod b4 ) ) & ( for b3, b4 being Element of Morphs c1 holds
( [b3,b4] in dom b2 implies b2 . [b3,b4] = b3 * b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def35 defines comp GRCAT_1:def 35 :
definition
let c
1 be
Universe;
func GroupCat c
1 -> CatStr equals :: GRCAT_1:def 36
CatStr(#
(GroupObjects a1),
(Morphs (GroupObjects a1)),
(dom (GroupObjects a1)),
(cod (GroupObjects a1)),
(comp (GroupObjects a1)),
(ID (GroupObjects a1)) #);
coherence
CatStr(# (GroupObjects c1),(Morphs (GroupObjects c1)),(dom (GroupObjects c1)),(cod (GroupObjects c1)),(comp (GroupObjects c1)),(ID (GroupObjects c1)) #) is CatStr
;
end;
:: deftheorem Def36 defines GroupCat GRCAT_1:def 36 :
theorem Th46: :: GRCAT_1:46
theorem Th47: :: GRCAT_1:47
theorem Th48: :: GRCAT_1:48
theorem Th49: :: GRCAT_1:49
theorem Th50: :: GRCAT_1:50
Lemma58:
for b1 being Universe
for b2, b3 being Morphism of (GroupCat b1) holds
( dom b3 = cod b2 implies ( dom (b3 * b2) = dom b2 & cod (b3 * b2) = cod b3 ) )
Lemma59:
for b1 being Universe
for b2, b3, b4 being Morphism of (GroupCat b1) holds
( dom b4 = cod b3 & dom b3 = cod b2 implies b4 * (b3 * b2) = (b4 * b3) * b2 )
Lemma60:
for b1 being Universe
for b2 being Object of (GroupCat b1) holds
( dom (id b2) = b2 & cod (id b2) = b2 & ( for b3 being Morphism of (GroupCat b1) holds
( cod b3 = b2 implies (id b2) * b3 = b3 ) ) & ( for b3 being Morphism of (GroupCat b1) holds
( dom b3 = b2 implies b3 * (id b2) = b3 ) ) )
:: deftheorem Def37 defines AbGroupObjects GRCAT_1:def 37 :
theorem Th51: :: GRCAT_1:51
:: deftheorem Def38 defines AbGroupCat GRCAT_1:def 38 :
theorem Th52: :: GRCAT_1:52
:: deftheorem Def39 defines MidOpGroupObjects GRCAT_1:def 39 :
:: deftheorem Def40 defines MidOpGroupCat GRCAT_1:def 40 :
theorem Th53: :: GRCAT_1:53
theorem Th54: :: GRCAT_1:54
theorem Th55: :: GRCAT_1:55
theorem Th56: :: GRCAT_1:56