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Theorem comfeqval 17751
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 7448 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩(compf𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf𝐷)𝑍))
32oveqd 7448 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹))
4 eqid 2737 . . 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 17743 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
14 eqid 2737 . . 3 (compf𝐷) = (compf𝐷)
15 eqid 2737 . . 3 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2737 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
17 comfeqval.2 . . 3 = (comp‘𝐷)
18 comfeqval.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
1918homfeqbas 17739 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
205, 19eqtrid 2789 . . . 4 (𝜑𝐵 = (Base‘𝐷))
218, 20eleqtrd 2843 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
229, 20eleqtrd 2843 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
2310, 20eleqtrd 2843 . . 3 (𝜑𝑍 ∈ (Base‘𝐷))
245, 6, 16, 18, 8, 9homfeqval 17740 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
2511, 24eleqtrd 2843 . . 3 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
265, 6, 16, 18, 9, 10homfeqval 17740 . . . 4 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
2712, 26eleqtrd 2843 . . 3 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 17743 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
293, 13, 283eqtr3d 2785 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cop 4632  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Homf chomf 17709  compfccomf 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-homf 17713  df-comf 17714
This theorem is referenced by:  catpropd  17752  cidpropd  17753  oppccomfpropd  17770  monpropd  17781  funcpropd  17947  natpropd  18024  fucpropd  18025  xpcpropd  18253  hofpropd  18312
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