:: OPPCAT_1 semantic presentation
theorem Th1: :: OPPCAT_1:1
definition
let C be
Category;
func c1 opp -> strict Category equals :: OPPCAT_1:def 1
CatStr(# the
Objects of
C,the
Morphisms of
C,the
Cod of
C,the
Dom of
C,
(~ the Comp of C),the
Id of
C #);
coherence
CatStr(# the Objects of C,the Morphisms of C,the Cod of C,the Dom of C,(~ the Comp of C),the Id of C #) is strict Category
by ;
end;
:: deftheorem Def1 defines opp OPPCAT_1:def 1 :
theorem Th2: :: OPPCAT_1:2
:: deftheorem Def2 defines opp OPPCAT_1:def 2 :
:: deftheorem Def3 defines opp OPPCAT_1:def 3 :
theorem Th3: :: OPPCAT_1:3
theorem Th4: :: OPPCAT_1:4
theorem Th5: :: OPPCAT_1:5
:: deftheorem Def4 defines opp OPPCAT_1:def 4 :
:: deftheorem Def5 defines opp OPPCAT_1:def 5 :
theorem Th6: :: OPPCAT_1:6
theorem Th7: :: OPPCAT_1:7
theorem Th8: :: OPPCAT_1:8
theorem Th9: :: OPPCAT_1:9
theorem Th10: :: OPPCAT_1:10
theorem Th11: :: OPPCAT_1:11
theorem Th12: :: OPPCAT_1:12
theorem Th13: :: OPPCAT_1:13
theorem Th14: :: OPPCAT_1:14
theorem Th15: :: OPPCAT_1:15
theorem Th16: :: OPPCAT_1:16
theorem Th17: :: OPPCAT_1:17
theorem Th18: :: OPPCAT_1:18
theorem Th19: :: OPPCAT_1:19
theorem Th20: :: OPPCAT_1:20
theorem Th21: :: OPPCAT_1:21
theorem Th22: :: OPPCAT_1:22
theorem Th23: :: OPPCAT_1:23
theorem Th24: :: OPPCAT_1:24
theorem Th25: :: OPPCAT_1:25
theorem Th26: :: OPPCAT_1:26
theorem Th27: :: OPPCAT_1:27
:: deftheorem Def6 defines /* OPPCAT_1:def 6 :
theorem Th28: :: OPPCAT_1:28
Lemma58:
for C, B being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp ))
theorem Th29: :: OPPCAT_1:29
theorem Th30: :: OPPCAT_1:30
Lemma60:
for C, B being Category
for S being Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)
Lemma62:
for C, B being Category
for S being Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = cod ((/* S) . f) & (Obj (/* S)) . (cod f) = dom ((/* S) . f) )
Lemma64:
for C, B being Category
for S being Functor of C opp ,B
for f, g being Morphism of C st dom g = cod f holds
(/* S) . (g * f) = ((/* S) . f) * ((/* S) . g)
:: deftheorem Def7 defines Contravariant_Functor OPPCAT_1:def 7 :
theorem Th31: :: OPPCAT_1:31
theorem Th32: :: OPPCAT_1:32
theorem Th33: :: OPPCAT_1:33
theorem Th34: :: OPPCAT_1:34
theorem Th35: :: OPPCAT_1:35
theorem Th36: :: OPPCAT_1:36
Lemma75:
for C, B being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj S) . (c opp ))
theorem Th37: :: OPPCAT_1:37
theorem Th38: :: OPPCAT_1:38
Lemma77:
for C, B being Category
for S being Contravariant_Functor of C opp ,B
for c being Object of C holds (/* S) . (id c) = id ((Obj (/* S)) . c)
Lemma78:
for C, B being Category
for S being Contravariant_Functor of C opp ,B
for f being Morphism of C holds
( (Obj (/* S)) . (dom f) = dom ((/* S) . f) & (Obj (/* S)) . (cod f) = cod ((/* S) . f) )
theorem Th39: :: OPPCAT_1:39
:: deftheorem Def8 defines *' OPPCAT_1:def 8 :
:: deftheorem Def9 defines *' OPPCAT_1:def 9 :
theorem Th40: :: OPPCAT_1:40
Lemma81:
for C, B being Category
for S being Functor of C,B
for c being Object of (C opp ) holds (*' S) . (id c) = id ((Obj S) . (opp c))
theorem Th41: :: OPPCAT_1:41
theorem Th42: :: OPPCAT_1:42
Lemma83:
for C, B being Category
for S being Functor of C,B
for c being Object of C holds (S *' ) . (id c) = id (((Obj S) . c) opp )
theorem Th43: :: OPPCAT_1:43
Lemma85:
for C, B being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp ) holds (*' S) . (id c) = id ((Obj S) . (opp c))
theorem Th44: :: OPPCAT_1:44
theorem Th45: :: OPPCAT_1:45
Lemma87:
for C, B being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *' ) . (id c) = id (((Obj S) . c) opp )
theorem Th46: :: OPPCAT_1:46
Lemma89:
for C, D being Category
for F being Function of the Morphisms of C,the Morphisms of D
for f being Morphism of (C opp ) holds ((*' F) *' ) . f = (F . (opp f)) opp
theorem Th47: :: OPPCAT_1:47
theorem Th48: :: OPPCAT_1:48
theorem Th49: :: OPPCAT_1:49
theorem Th50: :: OPPCAT_1:50
theorem Th51: :: OPPCAT_1:51
theorem Th52: :: OPPCAT_1:52
Lemma92:
for C, B, D being Category
for S being Function of the Morphisms of (C opp ),the Morphisms of B
for T being Function of the Morphisms of B,the Morphisms of D holds /* (T * S) = T * (/* S)
theorem Th53: :: OPPCAT_1:53
theorem Th54: :: OPPCAT_1:54
theorem Th55: :: OPPCAT_1:55
Lemma95:
for C, B being Category
for S being Contravariant_Functor of C,B
for c being Object of (C opp ) holds (*' S) . (id c) = id ((Obj (*' S)) . c)
Lemma96:
for C, B being Category
for S being Contravariant_Functor of C,B
for f being Morphism of (C opp ) holds
( (Obj (*' S)) . (dom f) = dom ((*' S) . f) & (Obj (*' S)) . (cod f) = cod ((*' S) . f) )
theorem Th56: :: OPPCAT_1:56
Lemma98:
for C, B being Category
for S being Contravariant_Functor of C,B
for c being Object of C holds (S *' ) . (id c) = id ((Obj (S *' )) . c)
Lemma99:
for C, B being Category
for S being Contravariant_Functor of C,B
for f being Morphism of C holds
( (Obj (S *' )) . (dom f) = dom ((S *' ) . f) & (Obj (S *' )) . (cod f) = cod ((S *' ) . f) )
theorem Th57: :: OPPCAT_1:57
Lemma101:
for C, B being Category
for S being Functor of C,B
for c being Object of (C opp ) holds (*' S) . (id c) = id ((Obj (*' S)) . c)
Lemma102:
for C, B being Category
for S being Functor of C,B
for f being Morphism of (C opp ) holds
( (Obj (*' S)) . (dom f) = cod ((*' S) . f) & (Obj (*' S)) . (cod f) = dom ((*' S) . f) )
theorem Th58: :: OPPCAT_1:58
Lemma104:
for C, B being Category
for S being Functor of C,B
for c being Object of C holds (S *' ) . (id c) = id ((Obj (S *' )) . c)
Lemma105:
for C, B being Category
for S being Functor of C,B
for f being Morphism of C holds
( (Obj (S *' )) . (dom f) = cod ((S *' ) . f) & (Obj (S *' )) . (cod f) = dom ((S *' ) . f) )
theorem Th59: :: OPPCAT_1:59
theorem Th60: :: OPPCAT_1:60
theorem Th61: :: OPPCAT_1:61
theorem Th62: :: OPPCAT_1:62
theorem Th63: :: OPPCAT_1:63
theorem Th64: :: OPPCAT_1:64
theorem Th65: :: OPPCAT_1:65
:: deftheorem Def10 defines id* OPPCAT_1:def 10 :
:: deftheorem Def11 defines *id OPPCAT_1:def 11 :
theorem Th66: :: OPPCAT_1:66
theorem Th67: :: OPPCAT_1:67
theorem Th68: :: OPPCAT_1:68
theorem Th69: :: OPPCAT_1:69
theorem Th70: :: OPPCAT_1:70