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Theorem sectmon 14708
Description: If  F is a section of  G, then  F is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (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 )
sectmon.1  |-  ( ph  ->  F ( X S Y ) G )
Assertion
Ref Expression
sectmon  |-  ( ph  ->  F  e.  ( X M Y ) )

Proof of Theorem sectmon
Dummy variables  g  h  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sectmon.1 . . . 4  |-  ( ph  ->  F ( X S Y ) G )
2 sectmon.b . . . . 5  |-  B  =  ( Base `  C
)
3 eqid 2438 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
4 eqid 2438 . . . . 5  |-  (comp `  C )  =  (comp `  C )
5 eqid 2438 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
6 sectmon.s . . . . 5  |-  S  =  (Sect `  C )
7 sectmon.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
8 sectmon.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 sectmon.y . . . . 5  |-  ( ph  ->  Y  e.  B )
102, 3, 4, 5, 6, 7, 8, 9issect 14684 . . . 4  |-  ( ph  ->  ( F ( X S Y ) G  <-> 
( F  e.  ( X ( Hom  `  C
) Y )  /\  G  e.  ( Y
( Hom  `  C ) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) ) ) )
111, 10mpbid 210 . . 3  |-  ( ph  ->  ( F  e.  ( X ( Hom  `  C
) Y )  /\  G  e.  ( Y
( Hom  `  C ) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) ) )
1211simp1d 1000 . 2  |-  ( ph  ->  F  e.  ( X ( Hom  `  C
) Y ) )
13 oveq2 6094 . . . . 5  |-  ( ( F ( <. x ,  X >. (comp `  C
) Y ) g )  =  ( F ( <. x ,  X >. (comp `  C ) Y ) h )  ->  ( G (
<. x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) g ) )  =  ( G ( <.
x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) h ) ) )
1411simp3d 1002 . . . . . . . . 9  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
1514ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( G (
<. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
1615oveq1d 6101 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( G ( <. X ,  Y >. (comp `  C ) X ) F ) ( <. x ,  X >. (comp `  C ) X ) g )  =  ( ( ( Id `  C ) `
 X ) (
<. x ,  X >. (comp `  C ) X ) g ) )
177ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  C  e.  Cat )
18 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  x  e.  B
)
198ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  X  e.  B
)
209ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  Y  e.  B
)
21 simprl 755 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  g  e.  ( x ( Hom  `  C
) X ) )
2212ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  F  e.  ( X ( Hom  `  C
) Y ) )
2311simp2d 1001 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( Y ( Hom  `  C
) X ) )
2423ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  G  e.  ( Y ( Hom  `  C
) X ) )
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 14616 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( G ( <. X ,  Y >. (comp `  C ) X ) F ) ( <. x ,  X >. (comp `  C ) X ) g )  =  ( G (
<. x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) g ) ) )
262, 3, 5, 17, 18, 4, 19, 21catlid 14613 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( ( Id `  C ) `
 X ) (
<. x ,  X >. (comp `  C ) X ) g )  =  g )
2716, 25, 263eqtr3d 2478 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( G (
<. x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) g ) )  =  g )
2815oveq1d 6101 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( G ( <. X ,  Y >. (comp `  C ) X ) F ) ( <. x ,  X >. (comp `  C ) X ) h )  =  ( ( ( Id `  C ) `
 X ) (
<. x ,  X >. (comp `  C ) X ) h ) )
29 simprr 756 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  h  e.  ( x ( Hom  `  C
) X ) )
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 14616 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( G ( <. X ,  Y >. (comp `  C ) X ) F ) ( <. x ,  X >. (comp `  C ) X ) h )  =  ( G (
<. x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) h ) ) )
312, 3, 5, 17, 18, 4, 19, 29catlid 14613 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( ( Id `  C ) `
 X ) (
<. x ,  X >. (comp `  C ) X ) h )  =  h )
3228, 30, 313eqtr3d 2478 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( G (
<. x ,  Y >. (comp `  C ) X ) ( F ( <.
x ,  X >. (comp `  C ) Y ) h ) )  =  h )
3327, 32eqeq12d 2452 . . . . 5  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( G ( <. x ,  Y >. (comp `  C ) X ) ( F ( <. x ,  X >. (comp `  C ) Y ) g ) )  =  ( G ( <. x ,  Y >. (comp `  C ) X ) ( F ( <. x ,  X >. (comp `  C ) Y ) h ) )  <->  g  =  h ) )
3413, 33syl5ib 219 . . . 4  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x ( Hom  `  C
) X )  /\  h  e.  ( x
( Hom  `  C ) X ) ) )  ->  ( ( F ( <. x ,  X >. (comp `  C ) Y ) g )  =  ( F (
<. x ,  X >. (comp `  C ) Y ) h )  ->  g  =  h ) )
3534ralrimivva 2803 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  A. g  e.  ( x ( Hom  `  C ) X ) A. h  e.  ( x ( Hom  `  C
) X ) ( ( F ( <.
x ,  X >. (comp `  C ) Y ) g )  =  ( F ( <. x ,  X >. (comp `  C
) Y ) h )  ->  g  =  h ) )
3635ralrimiva 2794 . 2  |-  ( ph  ->  A. x  e.  B  A. g  e.  (
x ( Hom  `  C
) X ) A. h  e.  ( x
( Hom  `  C ) X ) ( ( F ( <. x ,  X >. (comp `  C
) Y ) g )  =  ( F ( <. x ,  X >. (comp `  C ) Y ) h )  ->  g  =  h ) )
37 sectmon.m . . 3  |-  M  =  (Mono `  C )
382, 3, 4, 37, 7, 8, 9ismon2 14665 . 2  |-  ( ph  ->  ( F  e.  ( X M Y )  <-> 
( F  e.  ( X ( Hom  `  C
) Y )  /\  A. x  e.  B  A. g  e.  ( x
( Hom  `  C ) X ) A. h  e.  ( x ( Hom  `  C ) X ) ( ( F (
<. x ,  X >. (comp `  C ) Y ) g )  =  ( F ( <. x ,  X >. (comp `  C
) Y ) h )  ->  g  =  h ) ) ) )
3912, 36, 38mpbir2and 913 1  |-  ( ph  ->  F  e.  ( X M Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   <.cop 3878   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594   Idccid 14595  Monocmon 14659  Sectcsect 14675
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-cat 14598  df-cid 14599  df-mon 14661  df-sect 14678
This theorem is referenced by:  sectepi  14710
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