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Theorem monsect 15198
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 2457 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2457 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
5 eqid 2457 . . . . . . . . 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 15168 . . . . . . . 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 210 . . . . . . 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 1010 . . . . . 6  |-  ( ph  ->  ( F ( <. Y ,  X >. (comp `  C ) Y ) G )  =  ( ( Id `  C
) `  Y )
)
1312oveq1d 6311 . . . . 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 1009 . . . . . 6  |-  ( ph  ->  F  e.  ( X ( Hom  `  C
) Y ) )
1511simp1d 1008 . . . . . 6  |-  ( ph  ->  G  e.  ( Y ( Hom  `  C
) X ) )
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 15102 . . . . 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 15099 . . . . . 6  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  F )
182, 3, 5, 7, 9, 4, 8, 14catrid 15100 . . . . . 6  |-  ( ph  ->  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  F )
1917, 18eqtr4d 2501 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( F ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) )
2013, 16, 193eqtr3d 2506 . . . 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 15101 . . . . 5  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  e.  ( X ( Hom  `  C
) X ) )
242, 3, 5, 7, 9catidcl 15098 . . . . 5  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  C
) X ) )
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 15151 . . . 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 210 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 15169 . . 3  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
) )
2826, 27mpbird 232 . 2  |-  ( ph  ->  F ( X S Y ) G )
29 monsect.n . . 3  |-  N  =  (Inv `  C )
302, 29, 7, 9, 8, 6isinv 15175 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
3128, 1, 30mpbir2and 922 1  |-  ( ph  ->  F ( X N Y ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   Hom chom 14722  compcco 14723   Catccat 15080   Idccid 15081  Monocmon 15143  Sectcsect 15159  Invcinv 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-cat 15084  df-cid 15085  df-mon 15145  df-sect 15162  df-inv 15163
This theorem is referenced by:  episect  15200
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