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Theorem coa2 14216
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 14215 . . 3  |-  ( ph  ->  ( G  .x.  F
)  =  <. X ,  Z ,  ( ( 2nd `  G ) (
<. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)
76fveq2d 5724 . 2  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( 2nd `  <. X ,  Z ,  ( ( 2nd `  G
) ( <. X ,  Y >.  .xb  Z ) ( 2nd `  F ) ) >. ) )
8 ovex 6098 . . 3  |-  ( ( 2nd `  G ) ( <. X ,  Y >. 
.xb  Z ) ( 2nd `  F ) )  e.  _V
9 ot3rdg 6355 . . 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 8 . 2  |-  ( 2nd `  <. X ,  Z ,  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) >.
)  =  ( ( 2nd `  G ) ( <. X ,  Y >. 
.xb  Z ) ( 2nd `  F ) )
117, 10syl6eq 2483 1  |-  ( ph  ->  ( 2nd `  ( G  .x.  F ) )  =  ( ( 2nd `  G ) ( <. X ,  Y >.  .xb 
Z ) ( 2nd `  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   <.cotp 3810   ` cfv 5446  (class class class)co 6073   2ndc2nd 6340  compcco 13533  Homachoma 14170  compaccoa 14201
This theorem is referenced by:  arwass  14221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-doma 14171  df-coda 14172  df-homa 14173  df-arw 14174  df-coa 14203
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