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Theorem sectmon 13890
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 2366 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2366 . . . . 5  |-  (comp `  C )  =  (comp `  C )
5 eqid 2366 . . . . 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 13866 . . . 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 201 . . 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 968 . 2  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
13 oveq2 5989 . . . . 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 970 . . . . . . . . 9  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
1514ad2antrr 706 . . . . . . . 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 5996 . . . . . . 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 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  C  e.  Cat )
18 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  x  e.  B
)
198ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  X  e.  B
)
209ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  Y  e.  B
)
21 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  g  e.  ( x (  Hom  `  C
) X ) )
2212ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
2311simp2d 969 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
2423ad2antrr 706 . . . . . . . 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 13798 . . . . . . 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 13795 . . . . . . 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 2406 . . . . . 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 5996 . . . . . . 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 733 . . . . . . . 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 13798 . . . . . . 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 13795 . . . . . . 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 2406 . . . . . 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 2380 . . . . 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 210 . . . 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 2720 . . 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 2711 . 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 13847 . 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 888 1  |-  ( ph  ->  F  e.  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   A.wral 2628   <.cop 3732   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356    Hom chom 13427  compcco 13428   Catccat 13776   Idccid 13777  Monocmon 13841  Sectcsect 13857
This theorem is referenced by:  sectepi  13892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-cat 13780  df-cid 13781  df-mon 13843  df-sect 13860
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