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Theorem monsect 14722
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 2443 . . . . . . . . 9  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2443 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
5 eqid 2443 . . . . . . . . 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 14697 . . . . . . . 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 1002 . . . . . 6  |-  ( ph  ->  ( F ( <. Y ,  X >. (comp `  C ) Y ) G )  =  ( ( Id `  C
) `  Y )
)
1312oveq1d 6111 . . . . 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 1001 . . . . . 6  |-  ( ph  ->  F  e.  ( X ( Hom  `  C
) Y ) )
1511simp1d 1000 . . . . . 6  |-  ( ph  ->  G  e.  ( Y ( Hom  `  C
) X ) )
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 14629 . . . . 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 14626 . . . . . 6  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  F )
182, 3, 5, 7, 9, 4, 8, 14catrid 14627 . . . . . 6  |-  ( ph  ->  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  F )
1917, 18eqtr4d 2478 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( F ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) )
2013, 16, 193eqtr3d 2483 . . . 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 14628 . . . . 5  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  e.  ( X ( Hom  `  C
) X ) )
242, 3, 5, 7, 9catidcl 14625 . . . . 5  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  C
) X ) )
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 14680 . . . 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 14698 . . 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 14703 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
3128, 1, 30mpbir2and 913 1  |-  ( ph  ->  F ( X N Y ) G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   <.cop 3888   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   Hom chom 14254  compcco 14255   Catccat 14607   Idccid 14608  Monocmon 14672  Sectcsect 14688  Invcinv 14689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-cat 14611  df-cid 14612  df-mon 14674  df-sect 14691  df-inv 14692
This theorem is referenced by:  episect  14724
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