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Theorem coaval 14150
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 2387 . . 3  |-  (Nat `  C )  =  (Nat
`  C )
3 coaval.x . . 3  |-  .xb  =  (comp `  C )
41, 2, 3coafval 14146 . 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 14128 . . . 4  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3289 . . 3  |-  ( ph  ->  G  e.  (Nat `  C ) )
92, 5homarw 14128 . . . . 5  |-  ( X H Y )  C_  (Nat `  C )
10 homdmcoa.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
1110adantr 452 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  F  e.  ( X H Y ) )
129, 11sseldi 3289 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  F  e.  (Nat `  C )
)
135homacd 14123 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
1411, 13syl 16 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  Y )
15 simpr 448 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
1615fveq2d 5672 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  (domA `  G ) )
177adantr 452 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  G  e.  ( Y H Z ) )
185homadm 14122 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
1917, 18syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  G )  =  Y )
2016, 19eqtrd 2419 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  Y )
2114, 20eqtr4d 2422 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  (domA `  g ) )
22 fveq2 5668 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
2322eqeq1d 2395 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  F )  =  (domA `  g ) ) )
2423elrab 3035 . . . 4  |-  ( F  e.  { h  e.  (Nat `  C )  |  (coda
`  h )  =  (domA `  g ) }  <->  ( F  e.  (Nat `  C )  /\  (coda
`  F )  =  (domA `  g ) ) )
2512, 21, 24sylanbrc 646 . . 3  |-  ( (
ph  /\  g  =  G )  ->  F  e.  { h  e.  (Nat
`  C )  |  (coda
`  h )  =  (domA `  g ) } )
26 otex 4369 . . . 4  |-  <. (domA `  f ) ,  (coda
`  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.  e.  _V
2726a1i 11 . . 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 734 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
2928fveq2d 5672 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  (domA `  F
) )
305homadm 14122 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
3111, 30syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  F )  =  X )
3231adantrr 698 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  F )  =  X )
3329, 32eqtrd 2419 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  X )
3415fveq2d 5672 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  (coda `  G
) )
355homacd 14123 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (coda `  G
)  =  Z )
3617, 35syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  G
)  =  Z )
3734, 36eqtrd 2419 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  Z )
3837adantrr 698 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(coda `  g )  =  Z )
3920adantrr 698 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  g )  =  Y )
4033, 39opeq12d 3934 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (domA `  g
) >.  =  <. X ,  Y >. )
4140, 38oveq12d 6038 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( <. (domA `  f ) ,  (domA `  g
) >.  .xb  (coda
`  g ) )  =  ( <. X ,  Y >.  .xb  Z ) )
42 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
4342fveq2d 5672 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
4428fveq2d 5672 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
4541, 43, 44oveq123d 6041 . . . 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 3941 . . 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 6146 . 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 17 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 359    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899   <.cop 3760   <.cotp 3761   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   2ndc2nd 6287  compcco 13468  domAcdoma 14102  codaccoda 14103  Natcarw 14104  Homachoma 14105  compaccoa 14136
This theorem is referenced by:  coa2  14151  coahom  14152  arwlid  14154  arwrid  14155  arwass  14156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-ot 3767  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-doma 14106  df-coda 14107  df-homa 14108  df-arw 14109  df-coa 14138
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