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Theorem coaval 15058
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 2454 . . 3  |-  (Nat `  C )  =  (Nat
`  C )
3 coaval.x . . 3  |-  .xb  =  (comp `  C )
41, 2, 3coafval 15054 . 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 15036 . . . 4  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3465 . . 3  |-  ( ph  ->  G  e.  (Nat `  C ) )
92, 5homarw 15036 . . . . 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 3465 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  F  e.  (Nat `  C )
)
135homacd 15031 . . . . . 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 5806 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  (domA `  G ) )
177adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  G  e.  ( Y H Z ) )
185homadm 15030 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
1917, 18syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  G )  =  Y )
2016, 19eqtrd 2495 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  Y )
2114, 20eqtr4d 2498 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  (domA `  g ) )
22 fveq2 5802 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
2322eqeq1d 2456 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  F )  =  (domA `  g ) ) )
2423elrab 3224 . . . 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 4668 . . . 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 5806 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  (domA `  F
) )
305homadm 15030 . . . . . . 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 2495 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  X )
3415fveq2d 5806 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  (coda `  G
) )
355homacd 15031 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (coda `  G
)  =  Z )
3617, 35syl 16 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  G
)  =  Z )
3734, 36eqtrd 2495 . . . . 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 4178 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (domA `  g
) >.  =  <. X ,  Y >. )
4140, 38oveq12d 6221 . . . . 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 5806 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
4428fveq2d 5806 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
4541, 43, 44oveq123d 6224 . . . 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 4185 . . 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 6337 . 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 1370    e. wcel 1758   {crab 2803   _Vcvv 3078   <.cop 3994   <.cotp 3996   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   2ndc2nd 6689  compcco 14372  domAcdoma 15010  codaccoda 15011  Natcarw 15012  Homachoma 15013  compaccoa 15044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-doma 15014  df-coda 15015  df-homa 15016  df-arw 15017  df-coa 15046
This theorem is referenced by:  coa2  15059  coahom  15060  arwlid  15062  arwrid  15063  arwass  15064
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