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 → (Homf ‘𝐶) = (Homf ‘𝐷))
comfeqval.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
comfeqval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comfeqval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comfeqval.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
comfeqval.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
comfeqval.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))

Assertion

Ref	Expression
comfeqval	⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

Proof of Theorem comfeqval

Step	Hyp	Ref	Expression
1		comfeqval.4	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
2	1	oveqd 7448	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍))
3	2	oveqd 7448	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹))
4		eqid 2737	. . 3 ⊢ (compf‘𝐶) = (compf‘𝐶)
5		comfeqval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
6		comfeqval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
7		comfeqval.1	. . 3 ⊢ · = (comp‘𝐶)
8		comfeqval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		comfeqval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		comfeqval.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11		comfeqval.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
12		comfeqval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
13	4, 5, 6, 7, 8, 9, 10, 11, 12	comfval 17743	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
14		eqid 2737	. . 3 ⊢ (compf‘𝐷) = (compf‘𝐷)
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 → (Homf ‘𝐶) = (Homf ‘𝐷))
19	18	homfeqbas 17739	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
20	5, 19	eqtrid 2789	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
21	8, 20	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
22	9, 20	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	10, 20	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷))
24	5, 6, 16, 18, 8, 9	homfeqval 17740	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
25	11, 24	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
26	5, 6, 16, 18, 9, 10	homfeqval 17740	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
27	12, 26	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 17743	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
29	3, 13, 28	3eqtr3d 2785	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ⟨cop 4632  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Hom_f chomf 17709  comp_fccomf 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