MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coaval Unicode version

Theorem coaval 13900
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 2283 . . 3  |-  (Nat `  C )  =  (Nat
`  C )
3 coaval.x . . 3  |-  .xb  =  (comp `  C )
41, 2, 3coafval 13896 . 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 13878 . . . 4  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3178 . . 3  |-  ( ph  ->  G  e.  (Nat `  C ) )
92, 5homarw 13878 . . . . 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 3178 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  F  e.  (Nat `  C )
)
135homacd 13873 . . . . . 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 5529 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  (domA `  G ) )
177adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  =  G )  ->  G  e.  ( Y H Z ) )
185homadm 13872 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
1917, 18syl 15 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (domA `  G )  =  Y )
2016, 19eqtrd 2315 . . . . 5  |-  ( (
ph  /\  g  =  G )  ->  (domA `  g )  =  Y )
2114, 20eqtr4d 2318 . . . 4  |-  ( (
ph  /\  g  =  G )  ->  (coda `  F
)  =  (domA `  g ) )
22 fveq2 5525 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
2322eqeq1d 2291 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  F )  =  (domA `  g ) ) )
2423elrab 2923 . . . 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 4238 . . . 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 5529 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  (domA `  F
) )
305homadm 13872 . . . . . . 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 2315 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
(domA `  f )  =  X )
3415fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  g
)  =  (coda `  G
) )
355homacd 13873 . . . . . . 7  |-  ( G  e.  ( Y H Z )  ->  (coda `  G
)  =  Z )
3617, 35syl 15 . . . . . 6  |-  ( (
ph  /\  g  =  G )  ->  (coda `  G
)  =  Z )
3734, 36eqtrd 2315 . . . . 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 3804 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (domA `  f ) ,  (domA `  g
) >.  =  <. X ,  Y >. )
4140, 38oveq12d 5876 . . . . 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 5529 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
4428fveq2d 5529 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
4541, 43, 44oveq123d 5879 . . . 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 3811 . . 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 5981 . 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   <.cop 3643   <.cotp 3644   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121  compcco 13220  domAcdoma 13852  codaccoda 13853  Natcarw 13854  Homachoma 13855  compaccoa 13886
This theorem is referenced by:  coa2  13901  coahom  13902  arwlid  13904  arwrid  13905  arwass  13906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859  df-coa 13888
  Copyright terms: Public domain W3C validator