:: MODCAT_1 semantic presentation
:: deftheorem Def1 defines LeftMod_DOMAIN MODCAT_1:def 1 :
:: deftheorem Def2 defines LModMorphism_DOMAIN MODCAT_1:def 2 :
theorem Th1: :: MODCAT_1:1
canceled;
theorem Th2: :: MODCAT_1:2
canceled;
theorem Th3: :: MODCAT_1:3
:: deftheorem Def3 defines LModMorphism_DOMAIN MODCAT_1:def 3 :
theorem Th4: :: MODCAT_1:4
theorem Th5: :: MODCAT_1:5
definition
let c
1 be
Ring;
let c
2, c
3 be
LeftMod of c
1;
func Morphs c
2,c
3 -> LModMorphism_DOMAIN of a
2,a
3 means :
Def4:
:: MODCAT_1:def 4
for b
1 being
set holds
( b
1 in a
4 iff b
1 is
strict Morphism of a
2,a
3 );
existence
ex b1 being LModMorphism_DOMAIN of c2,c3 st
for b2 being set holds
( b2 in b1 iff b2 is strict Morphism of c2,c3 )
uniqueness
for b1, b2 being LModMorphism_DOMAIN of c2,c3 holds
( ( for b3 being set holds
( b3 in b1 iff b3 is strict Morphism of c2,c3 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is strict Morphism of c2,c3 ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines Morphs MODCAT_1:def 4 :
:: deftheorem Def5 defines GO MODCAT_1:def 5 :
theorem Th6: :: MODCAT_1:6
for b
1 being
Ringfor b
2, b
3, b
4 being
set holds
(
GO b
2,b
3,b
1 &
GO b
2,b
4,b
1 implies b
3 = b
4 )
theorem Th7: :: MODCAT_1:7
definition
let c
1 be
Universe;
let c
2 be
Ring;
func LModObjects c
1,c
2 -> set means :
Def6:
:: MODCAT_1:def 6
for b
1 being
set holds
( b
1 in a
3 iff ex b
2 being
set st
( b
2 in { [b3,b4] where B is Element of GroupObjects a1, B is Element of Funcs [:the carrier of a2,{{} }:],{{} } : verum } &
GO b
2,b
1,a
2 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ex b3 being set st
( b3 in { [b4,b5] where B is Element of GroupObjects c1, B is Element of Funcs [:the carrier of c2,{{} }:],{{} } : verum } & GO b3,b2,c2 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ex b4 being set st
( b4 in { [b5,b6] where B is Element of GroupObjects c1, B is Element of Funcs [:the carrier of c2,{{} }:],{{} } : verum } & GO b4,b3,c2 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4 being set st
( b4 in { [b5,b6] where B is Element of GroupObjects c1, B is Element of Funcs [:the carrier of c2,{{} }:],{{} } : verum } & GO b4,b3,c2 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines LModObjects MODCAT_1:def 6 :
theorem Th8: :: MODCAT_1:8
theorem Th9: :: MODCAT_1:9
:: deftheorem Def7 defines Morphs MODCAT_1:def 7 :
:: deftheorem Def8 defines dom' MODCAT_1:def 8 :
:: deftheorem Def9 defines cod' MODCAT_1:def 9 :
:: deftheorem Def10 defines ID MODCAT_1:def 10 :
definition
let c
1 be
Ring;
let c
2 be
LeftMod_DOMAIN of c
1;
func dom c
2 -> Function of
Morphs a
2,a
2 means :
Def11:
:: MODCAT_1:def 11
for b
1 being
Element of
Morphs a
2 holds a
3 . b
1 = dom' b
1;
existence
ex b1 being Function of Morphs c2,c2 st
for b2 being Element of Morphs c2 holds b1 . b2 = dom' b2
uniqueness
for b1, b2 being Function of Morphs c2,c2 holds
( ( for b3 being Element of Morphs c2 holds b1 . b3 = dom' b3 ) & ( for b3 being Element of Morphs c2 holds b2 . b3 = dom' b3 ) implies b1 = b2 )
func cod c
2 -> Function of
Morphs a
2,a
2 means :
Def12:
:: MODCAT_1:def 12
for b
1 being
Element of
Morphs a
2 holds a
3 . b
1 = cod' b
1;
existence
ex b1 being Function of Morphs c2,c2 st
for b2 being Element of Morphs c2 holds b1 . b2 = cod' b2
uniqueness
for b1, b2 being Function of Morphs c2,c2 holds
( ( for b3 being Element of Morphs c2 holds b1 . b3 = cod' b3 ) & ( for b3 being Element of Morphs c2 holds b2 . b3 = cod' b3 ) implies b1 = b2 )
func ID c
2 -> Function of a
2,
Morphs a
2 means :
Def13:
:: MODCAT_1:def 13
for b
1 being
Element of a
2 holds a
3 . b
1 = ID b
1;
existence
ex b1 being Function of c2, Morphs c2 st
for b2 being Element of c2 holds b1 . b2 = ID b2
uniqueness
for b1, b2 being Function of c2, Morphs c2 holds
( ( for b3 being Element of c2 holds b1 . b3 = ID b3 ) & ( for b3 being Element of c2 holds b2 . b3 = ID b3 ) implies b1 = b2 )
end;
:: deftheorem Def11 defines dom MODCAT_1:def 11 :
:: deftheorem Def12 defines cod MODCAT_1:def 12 :
:: deftheorem Def13 defines ID MODCAT_1:def 13 :
theorem Th10: :: MODCAT_1:10
theorem Th11: :: MODCAT_1:11
theorem Th12: :: MODCAT_1:12
definition
let c
1 be
Ring;
let c
2 be
LeftMod_DOMAIN of c
1;
func comp c
2 -> PartFunc of
[:(Morphs a2),(Morphs a2):],
Morphs a
2 means :
Def14:
:: MODCAT_1:def 14
( ( for b
1, b
2 being
Element of
Morphs a
2 holds
(
[b1,b2] in dom a
3 iff
dom' b
1 = cod' b
2 ) ) & ( for b
1, b
2 being
Element of
Morphs a
2 holds
(
[b1,b2] in dom a
3 implies a
3 . [b1,b2] = b
1 * b
2 ) ) );
existence
ex b1 being PartFunc of [:(Morphs c2),(Morphs c2):], Morphs c2 st
( ( for b2, b3 being Element of Morphs c2 holds
( [b2,b3] in dom b1 iff dom' b2 = cod' b3 ) ) & ( for b2, b3 being Element of Morphs c2 holds
( [b2,b3] in dom b1 implies b1 . [b2,b3] = b2 * b3 ) ) )
uniqueness
for b1, b2 being PartFunc of [:(Morphs c2),(Morphs c2):], Morphs c2 holds
( ( for b3, b4 being Element of Morphs c2 holds
( [b3,b4] in dom b1 iff dom' b3 = cod' b4 ) ) & ( for b3, b4 being Element of Morphs c2 holds
( [b3,b4] in dom b1 implies b1 . [b3,b4] = b3 * b4 ) ) & ( for b3, b4 being Element of Morphs c2 holds
( [b3,b4] in dom b2 iff dom' b3 = cod' b4 ) ) & ( for b3, b4 being Element of Morphs c2 holds
( [b3,b4] in dom b2 implies b2 . [b3,b4] = b3 * b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def14 defines comp MODCAT_1:def 14 :
theorem Th13: :: MODCAT_1:13
definition
let c
1 be
Universe;
let c
2 be
Ring;
func LModCat c
1,c
2 -> strict CatStr equals :: MODCAT_1:def 15
CatStr(#
(LModObjects a1,a2),
(Morphs (LModObjects a1,a2)),
(dom (LModObjects a1,a2)),
(cod (LModObjects a1,a2)),
(comp (LModObjects a1,a2)),
(ID (LModObjects a1,a2)) #);
coherence
CatStr(# (LModObjects c1,c2),(Morphs (LModObjects c1,c2)),(dom (LModObjects c1,c2)),(cod (LModObjects c1,c2)),(comp (LModObjects c1,c2)),(ID (LModObjects c1,c2)) #) is strict CatStr
;
end;
:: deftheorem Def15 defines LModCat MODCAT_1:def 15 :
for b
1 being
Universefor b
2 being
Ring holds
LModCat b
1,b
2 = CatStr(#
(LModObjects b1,b2),
(Morphs (LModObjects b1,b2)),
(dom (LModObjects b1,b2)),
(cod (LModObjects b1,b2)),
(comp (LModObjects b1,b2)),
(ID (LModObjects b1,b2)) #);
theorem Th14: :: MODCAT_1:14
theorem Th15: :: MODCAT_1:15
theorem Th16: :: MODCAT_1:16
theorem Th17: :: MODCAT_1:17
theorem Th18: :: MODCAT_1:18
Lemma27:
for b1 being Universe
for b2 being Ring
for b3, b4 being Morphism of (LModCat b1,b2) holds
( dom b4 = cod b3 implies ( dom (b4 * b3) = dom b3 & cod (b4 * b3) = cod b4 ) )
Lemma28:
for b1 being Universe
for b2 being Ring
for b3, b4, b5 being Morphism of (LModCat b1,b2) holds
( dom b5 = cod b4 & dom b4 = cod b3 implies b5 * (b4 * b3) = (b5 * b4) * b3 )
Lemma29:
for b1 being Universe
for b2 being Ring
for b3 being Object of (LModCat b1,b2) holds
( dom (id b3) = b3 & cod (id b3) = b3 & ( for b4 being Morphism of (LModCat b1,b2) holds
( cod b4 = b3 implies (id b3) * b4 = b4 ) ) & ( for b4 being Morphism of (LModCat b1,b2) holds
( dom b4 = b3 implies b4 * (id b3) = b4 ) ) )