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Theorem monsect 14005
Description: If  F is a monomorphism and  G is a section of  F, then  G is an inverse of  F and they are both isomorphisms. This is also stated as "a monomorphism which is also a split epimorphism is an isomorphism". (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
sectmon.b  |-  B  =  ( Base `  C
)
sectmon.m  |-  M  =  (Mono `  C )
sectmon.s  |-  S  =  (Sect `  C )
sectmon.c  |-  ( ph  ->  C  e.  Cat )
sectmon.x  |-  ( ph  ->  X  e.  B )
sectmon.y  |-  ( ph  ->  Y  e.  B )
monsect.n  |-  N  =  (Inv `  C )
monsect.1  |-  ( ph  ->  F  e.  ( X M Y ) )
monsect.2  |-  ( ph  ->  G ( Y S X ) F )
Assertion
Ref Expression
monsect  |-  ( ph  ->  F ( X N Y ) G )

Proof of Theorem monsect
StepHypRef Expression
1 monsect.2 . . . . . . . 8  |-  ( ph  ->  G ( Y S X ) F )
2 sectmon.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
3 eqid 2437 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2437 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
5 eqid 2437 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
6 sectmon.s . . . . . . . . 9  |-  S  =  (Sect `  C )
7 sectmon.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
8 sectmon.y . . . . . . . . 9  |-  ( ph  ->  Y  e.  B )
9 sectmon.x . . . . . . . . 9  |-  ( ph  ->  X  e.  B )
102, 3, 4, 5, 6, 7, 8, 9issect 13980 . . . . . . . 8  |-  ( ph  ->  ( G ( Y S X ) F  <-> 
( G  e.  ( Y (  Hom  `  C
) X )  /\  F  e.  ( X
(  Hom  `  C ) Y )  /\  ( F ( <. Y ,  X >. (comp `  C
) Y ) G )  =  ( ( Id `  C ) `
 Y ) ) ) )
111, 10mpbid 203 . . . . . . 7  |-  ( ph  ->  ( G  e.  ( Y (  Hom  `  C
) X )  /\  F  e.  ( X
(  Hom  `  C ) Y )  /\  ( F ( <. Y ,  X >. (comp `  C
) Y ) G )  =  ( ( Id `  C ) `
 Y ) ) )
1211simp3d 972 . . . . . 6  |-  ( ph  ->  ( F ( <. Y ,  X >. (comp `  C ) Y ) G )  =  ( ( Id `  C
) `  Y )
)
1312oveq1d 6097 . . . . 5  |-  ( ph  ->  ( ( F (
<. Y ,  X >. (comp `  C ) Y ) G ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( ( ( Id `  C ) `  Y
) ( <. X ,  Y >. (comp `  C
) Y ) F ) )
1411simp2d 971 . . . . . 6  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
1511simp1d 970 . . . . . 6  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 13912 . . . . 5  |-  ( ph  ->  ( ( F (
<. Y ,  X >. (comp `  C ) Y ) G ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( F ( <. X ,  X >. (comp `  C
) Y ) ( G ( <. X ,  Y >. (comp `  C
) X ) F ) ) )
172, 3, 5, 7, 9, 4, 8, 14catlid 13909 . . . . . 6  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  F )
182, 3, 5, 7, 9, 4, 8, 14catrid 13910 . . . . . 6  |-  ( ph  ->  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  F )
1917, 18eqtr4d 2472 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( F ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) )
2013, 16, 193eqtr3d 2477 . . . 4  |-  ( ph  ->  ( F ( <. X ,  X >. (comp `  C ) Y ) ( G ( <. X ,  Y >. (comp `  C ) X ) F ) )  =  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) ) )
21 sectmon.m . . . . 5  |-  M  =  (Mono `  C )
22 monsect.1 . . . . 5  |-  ( ph  ->  F  e.  ( X M Y ) )
232, 3, 4, 7, 9, 8, 9, 14, 15catcocl 13911 . . . . 5  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  e.  ( X (  Hom  `  C
) X ) )
242, 3, 5, 7, 9catidcl 13908 . . . . 5  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X (  Hom  `  C
) X ) )
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 13963 . . . 4  |-  ( ph  ->  ( ( F (
<. X ,  X >. (comp `  C ) Y ) ( G ( <. X ,  Y >. (comp `  C ) X ) F ) )  =  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  <->  ( G
( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id
`  C ) `  X ) ) )
2620, 25mpbid 203 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 13981 . . 3  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
) )
2826, 27mpbird 225 . 2  |-  ( ph  ->  F ( X S Y ) G )
29 monsect.n . . 3  |-  N  =  (Inv `  C )
302, 29, 7, 9, 8, 6isinv 13986 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
3128, 1, 30mpbir2and 890 1  |-  ( ph  ->  F ( X N Y ) G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3818   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470    Hom chom 13541  compcco 13542   Catccat 13890   Idccid 13891  Monocmon 13955  Sectcsect 13971  Invcinv 13972
This theorem is referenced by:  episect  14007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-cat 13894  df-cid 13895  df-mon 13957  df-sect 13974  df-inv 13975
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