MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coa2 Structured version   Unicode version

Theorem coa2 15245
Description: The morphism part of arrow composition. (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
coa2  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) )

Proof of Theorem coa2
StepHypRef Expression
1 homdmcoa.o . . . 4  |-  .x.  =  (compa `  C )
2 homdmcoa.h . . . 4  |-  H  =  (Homa
`  C )
3 homdmcoa.f . . . 4  |-  ( ph  ->  F  e.  ( X H Y ) )
4 homdmcoa.g . . . 4  |-  ( ph  ->  G  e.  ( Y H Z ) )
5 coaval.x . . . 4  |-  .xb  =  (comp `  C )
61, 2, 3, 4, 5coaval 15244 . . 3  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)
76fveq2d 5863 . 2  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( 2nd `  <. X ,  Z ,  ( ( 2nd `  G
) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F ) ) >. ) )
8 ovex 6302 . . 3  |-  ( ( 2nd `  G ) ( <. X ,  Y >. 
.xb  Z ) ( 2nd `  F ) )  e.  _V
9 ot3rdg 6792 . . 3  |-  ( ( ( 2nd `  G
) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F ) )  e.  _V  ->  ( 2nd `  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)  =  ( ( 2nd `  G ) ( <. X ,  Y >. 
.xb  Z ) ( 2nd `  F ) ) )
108, 9ax-mp 5 . 2  |-  ( 2nd `  <. X ,  Z ,  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)  =  ( ( 2nd `  G ) ( <. X ,  Y >. 
.xb  Z ) ( 2nd `  F ) )
117, 10syl6eq 2519 1  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028   <.cotp 4030   ` cfv 5581  (class class class)co 6277   2ndc2nd 6775  compcco 14558  Homachoma 15199  compaccoa 15230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-ot 4031  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-doma 15200  df-coda 15201  df-homa 15202  df-arw 15203  df-coa 15232
This theorem is referenced by:  arwass  15250
  Copyright terms: Public domain W3C validator