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Theorem coaval 13916
Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o  |-  .x.  =  (compa `  C )
homdmcoa.h  |-  H  =  (Homa
`  C )
homdmcoa.f  |-  ( ph  ->  F  e.  ( X H Y ) )
homdmcoa.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
coaval.x  |-  .xb  =  (comp `  C )
Assertion
Ref Expression
coaval  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)

Proof of Theorem coaval
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homdmcoa.o . . 3  |-  .x.  =  (compa `  C )
2 eqid 2296 . . 3  |-  (Nat `  C )  =  (Nat
`  C )
3 coaval.x . . 3  |-  .xb  =  (comp `  C )
41, 2, 3coafval 13912 . 2  |-  .x.  =  ( g  e.  (Nat
`  C ) ,  f  e.  { h  e.  (Nat `  C )  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
5 homdmcoa.h . . . . 5  |-  H  =  (Homa
`  C )
62, 5homarw 13894 . . . 4  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3191 . . 3  |-  ( ph  ->  G  e.  (Nat `  C ) )
92, 5homarw 13894 . . . . 5  |-  ( X H Y )  C_  (Nat `  C )
10 homdmcoa.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  F  e.  ( X H Y ) )
129, 11sseldi 3191 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  F  e.  (Nat `  C )
)
135homacd 13889 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
1411, 13syl 15 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  Y )
15 simpr 447 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
1615fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  (domA `  G ) )
177adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  G  e.  ( Y H Z ) )
185homadm 13888 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
1917, 18syl 15 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  G )  =  Y )
2016, 19eqtrd 2328 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  Y )
2114, 20eqtr4d 2331 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  (domA `  g ) )
22 fveq2 5541 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
2322eqeq1d 2304 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  F )  =  (domA `  g ) ) )
2423elrab 2936 . . . 4  |-  ( F  e.  { h  e.  (Nat `  C )  |  (coda
`  h )  =  (domA `  g ) }  <->  ( F  e.  (Nat `  C )  /\  (coda
`  F )  =  (domA `  g ) ) )
2512, 21, 24sylanbrc 645 . . 3  |-  ( (
ph  /\  g  =  G )  ->  F  e.  { h  e.  (Nat
`  C )  |  (coda
`  h )  =  (domA `  g ) } )
26 otex 4254 . . . 4  |-  <. (domA `  f ) ,  (coda
`  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.  e.  _V
2726a1i 10 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.  e.  _V )
28 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
2928fveq2d 5545 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  (domA `  F
) )
305homadm 13888 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
3111, 30syl 15 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  F )  =  X )
3231adantrr 697 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  F )  =  X )
3329, 32eqtrd 2328 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  X )
3415fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  (coda `  G
) )
355homacd 13889 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (coda `  G
)  =  Z )
3617, 35syl 15 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  G
)  =  Z )
3734, 36eqtrd 2328 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  Z )
3837adantrr 697 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(coda `  g )  =  Z )
3920adantrr 697 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  g )  =  Y )
4033, 39opeq12d 3820 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (domA `  g
) >.  =  <. X ,  Y >. )
4140, 38oveq12d 5892 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( <. (domA `  f ) ,  (domA `  g
) >.  .xb  (coda
`  g ) )  =  ( <. X ,  Y >.  .xb  Z ) )
42 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
4342fveq2d 5545 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
4428fveq2d 5545 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
4541, 43, 44oveq123d 5895 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) )  =  ( ( 2nd `  G
) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F ) ) )
4633, 38, 45oteq123d 3827 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.  =  <. X ,  Z ,  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)
478, 25, 27, 46ovmpt2dv2 5997 . 2  |-  ( ph  ->  (  .x.  =  ( g  e.  (Nat `  C ) ,  f  e.  { h  e.  (Nat `  C )  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)  ->  ( G  .x.  F )  =  <. X ,  Z ,  ( ( 2nd `  G
) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F ) ) >. ) )
484, 47mpi 16 1  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   <.cop 3656   <.cotp 3657   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137  compcco 13236  domAcdoma 13868  codaccoda 13869  Natcarw 13870  Homachoma 13871  compaccoa 13902
This theorem is referenced by:  coa2  13917  coahom  13918  arwlid  13920  arwrid  13921  arwass  13922
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-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-ot 3663  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-doma 13872  df-coda 13873  df-homa 13874  df-arw 13875  df-coa 13904
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