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Theorem comfeqval 17721
Description: Equality of two compositions. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeqval.b 𝐵 = (Base‘𝐶)
comfeqval.h 𝐻 = (Hom ‘𝐶)
comfeqval.1 · = (comp‘𝐶)
comfeqval.2 = (comp‘𝐷)
comfeqval.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
comfeqval.4 (𝜑 → (compf𝐶) = (compf𝐷))
comfeqval.x (𝜑𝑋𝐵)
comfeqval.y (𝜑𝑌𝐵)
comfeqval.z (𝜑𝑍𝐵)
comfeqval.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
comfeqval.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
comfeqval (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))

Proof of Theorem comfeqval
StepHypRef Expression
1 comfeqval.4 . . . 4 (𝜑 → (compf𝐶) = (compf𝐷))
21oveqd 7441 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩(compf𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf𝐷)𝑍))
32oveqd 7441 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹))
4 eqid 2726 . . 3 (compf𝐶) = (compf𝐶)
5 comfeqval.b . . 3 𝐵 = (Base‘𝐶)
6 comfeqval.h . . 3 𝐻 = (Hom ‘𝐶)
7 comfeqval.1 . . 3 · = (comp‘𝐶)
8 comfeqval.x . . 3 (𝜑𝑋𝐵)
9 comfeqval.y . . 3 (𝜑𝑌𝐵)
10 comfeqval.z . . 3 (𝜑𝑍𝐵)
11 comfeqval.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
12 comfeqval.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
134, 5, 6, 7, 8, 9, 10, 11, 12comfval 17713 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
14 eqid 2726 . . 3 (compf𝐷) = (compf𝐷)
15 eqid 2726 . . 3 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2726 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
17 comfeqval.2 . . 3 = (comp‘𝐷)
18 comfeqval.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
1918homfeqbas 17709 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
205, 19eqtrid 2778 . . . 4 (𝜑𝐵 = (Base‘𝐷))
218, 20eleqtrd 2828 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
229, 20eleqtrd 2828 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
2310, 20eleqtrd 2828 . . 3 (𝜑𝑍 ∈ (Base‘𝐷))
245, 6, 16, 18, 8, 9homfeqval 17710 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
2511, 24eleqtrd 2828 . . 3 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
265, 6, 16, 18, 9, 10homfeqval 17710 . . . 4 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
2712, 26eleqtrd 2828 . . 3 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 17713 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
293, 13, 283eqtr3d 2774 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cop 4639  cfv 6554  (class class class)co 7424  Basecbs 17213  Hom chom 17277  compcco 17278  Homf chomf 17679  compfccomf 17680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-homf 17683  df-comf 17684
This theorem is referenced by:  catpropd  17722  cidpropd  17723  oppccomfpropd  17742  monpropd  17753  funcpropd  17922  natpropd  18001  fucpropd  18002  xpcpropd  18233  hofpropd  18292
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