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Theorem monsect 13697
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 2296 . . . . . . . . 9  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2296 . . . . . . . . 9  |-  (comp `  C )  =  (comp `  C )
5 eqid 2296 . . . . . . . . 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 13672 . . . . . . . 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 201 . . . . . . 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 969 . . . . . 6  |-  ( ph  ->  ( F ( <. Y ,  X >. (comp `  C ) Y ) G )  =  ( ( Id `  C
) `  Y )
)
1312oveq1d 5889 . . . . 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 968 . . . . . 6  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
1511simp1d 967 . . . . . 6  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
162, 3, 4, 7, 9, 8, 9, 14, 15, 8, 14catass 13604 . . . . 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 13601 . . . . . 6  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  F )
182, 3, 5, 7, 9, 4, 8, 14catrid 13602 . . . . . 6  |-  ( ph  ->  ( F ( <. X ,  X >. (comp `  C ) Y ) ( ( Id `  C ) `  X
) )  =  F )
1917, 18eqtr4d 2331 . . . . 5  |-  ( ph  ->  ( ( ( Id
`  C ) `  Y ) ( <. X ,  Y >. (comp `  C ) Y ) F )  =  ( F ( <. X ,  X >. (comp `  C
) Y ) ( ( Id `  C
) `  X )
) )
2013, 16, 193eqtr3d 2336 . . . 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 13603 . . . . 5  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  e.  ( X (  Hom  `  C
) X ) )
242, 3, 5, 7, 9catidcl 13600 . . . . 5  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X (  Hom  `  C
) X ) )
252, 3, 4, 21, 7, 9, 8, 9, 22, 23, 24moni 13655 . . . 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 201 . . 3  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
272, 3, 4, 5, 6, 7, 9, 8, 14, 15issect2 13673 . . 3  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
) )
2826, 27mpbird 223 . 2  |-  ( ph  ->  F ( X S Y ) G )
29 monsect.n . . 3  |-  N  =  (Inv `  C )
302, 29, 7, 9, 8, 6isinv 13678 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
3128, 1, 30mpbir2and 888 1  |-  ( ph  ->  F ( X N Y ) G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  Monocmon 13647  Sectcsect 13663  Invcinv 13664
This theorem is referenced by:  episect  13699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-cat 13586  df-cid 13587  df-mon 13649  df-sect 13666  df-inv 13667
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