Theorem comfeqval 17723

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 7409	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍))
3	2	oveqd 7409	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹))
4		eqid 2761	. . 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 17715	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
14		eqid 2761	. . 3 ⊢ (compf‘𝐷) = (compf‘𝐷)
15		eqid 2761	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2761	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		comfeqval.2	. . 3 ⊢ ∙ = (comp‘𝐷)
18		comfeqval.3	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
19	18	homfeqbas 17711	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
20	5, 19	eqtrid 2808	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
21	8, 20	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
22	9, 20	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	10, 20	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷))
24	5, 6, 16, 18, 8, 9	homfeqval 17712	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
25	11, 24	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
26	5, 6, 16, 18, 9, 10	homfeqval 17712	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
27	12, 26	eleqtrd 2863	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 17715	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
29	3, 13, 28	3eqtr3d 2804	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ⟨cop 4587  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  Hom chom 17280  compcco 17281  Hom_f chomf 17681  comp_fccomf 17682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-homf 17685  df-comf 17686

This theorem is referenced by:  catpropd  17724  cidpropd  17725  oppccomfpropd  17742  monpropd  17753  funcpropd  17918  natpropd  17995  fucpropd  17996  xpcpropd  18223  hofpropd  18282  sectpropdlem  49621  uppropd  49766