Theorem comfeqval 17643

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 7385	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍))
3	2	oveqd 7385	. 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 17635	. 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 17631	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
20	5, 19	eqtrid 2784	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
21	8, 20	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
22	9, 20	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	10, 20	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷))
24	5, 6, 16, 18, 8, 9	homfeqval 17632	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
25	11, 24	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
26	5, 6, 16, 18, 9, 10	homfeqval 17632	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
27	12, 26	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 17635	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
29	3, 13, 28	3eqtr3d 2780	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4588  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  compcco 17201  Hom_f chomf 17601  comp_fccomf 17602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-homf 17605  df-comf 17606

This theorem is referenced by:  catpropd  17644  cidpropd  17645  oppccomfpropd  17662  monpropd  17673  funcpropd  17838  natpropd  17915  fucpropd  17916  xpcpropd  18143  hofpropd  18202  sectpropdlem  49389  uppropd  49534