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Theorem coaval 15270
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 2467 . . 3  |-  (Nat `  C )  =  (Nat
`  C )
3 coaval.x . . 3  |-  .xb  =  (comp `  C )
41, 2, 3coafval 15266 . 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 15248 . . . 4  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3507 . . 3  |-  ( ph  ->  G  e.  (Nat `  C ) )
92, 5homarw 15248 . . . . 5  |-  ( X H Y )  C_  (Nat `  C )
10 homdmcoa.f . . . . . 6  |-  ( ph  ->  F  e.  ( X H Y ) )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  F  e.  ( X H Y ) )
129, 11sseldi 3507 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  F  e.  (Nat `  C )
)
135homacd 15243 . . . . . 6  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
1411, 13syl 16 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  Y )
15 simpr 461 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  g  =  G )
1615fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  (domA `  G ) )
177adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  G  e.  ( Y H Z ) )
185homadm 15242 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
1917, 18syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  G )  =  Y )
2016, 19eqtrd 2508 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  Y )
2114, 20eqtr4d 2511 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  (domA `  g ) )
22 fveq2 5872 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
2322eqeq1d 2469 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  F )  =  (domA `  g ) ) )
2423elrab 3266 . . . 4  |-  ( F  e.  { h  e.  (Nat `  C )  |  (coda
`  h )  =  (domA `  g ) }  <->  ( F  e.  (Nat `  C )  /\  (coda
`  F )  =  (domA `  g ) ) )
2512, 21, 24sylanbrc 664 . . 3  |-  ( (
ph  /\  g  =  G )  ->  F  e.  { h  e.  (Nat
`  C )  |  (coda
`  h )  =  (domA `  g ) } )
26 otex 4718 . . . 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 756 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
2928fveq2d 5876 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  (domA `  F
) )
305homadm 15242 . . . . . . 7  |-  ( F  e.  ( X H Y )  ->  (domA `  F )  =  X )
3111, 30syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  F )  =  X )
3231adantrr 716 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  F )  =  X )
3329, 32eqtrd 2508 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  X )
3415fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  (coda `  G
) )
355homacd 15243 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (coda `  G
)  =  Z )
3617, 35syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  G
)  =  Z )
3734, 36eqtrd 2508 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  Z )
3837adantrr 716 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(coda `  g )  =  Z )
3920adantrr 716 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  g )  =  Y )
4033, 39opeq12d 4227 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (domA `  g
) >.  =  <. X ,  Y >. )
4140, 38oveq12d 6313 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( <. (domA `  f ) ,  (domA `  g
) >.  .xb  (coda
`  g ) )  =  ( <. X ,  Y >.  .xb  Z ) )
42 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
4342fveq2d 5876 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
4428fveq2d 5876 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
4541, 43, 44oveq123d 6316 . . . 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 4234 . . 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 6431 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   <.cop 4039   <.cotp 4041   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   2ndc2nd 6794  compcco 14584  domAcdoma 15222  codaccoda 15223  Natcarw 15224  Homachoma 15225  compaccoa 15256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-doma 15226  df-coda 15227  df-homa 15228  df-arw 15229  df-coa 15258
This theorem is referenced by:  coa2  15271  coahom  15272  arwlid  15274  arwrid  15275  arwass  15276
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