Theorem comfeqval 17802

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 7434	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍))
3	2	oveqd 7434	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹))
4		eqid 2762	. . 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 17794	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
14		eqid 2762	. . 3 ⊢ (compf‘𝐷) = (compf‘𝐷)
15		eqid 2762	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2762	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		comfeqval.2	. . 3 ⊢ ∙ = (comp‘𝐷)
18		comfeqval.3	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
19	18	homfeqbas 17790	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
20	5, 19	eqtrid 2809	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
21	8, 20	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
22	9, 20	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	10, 20	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷))
24	5, 6, 16, 18, 8, 9	homfeqval 17791	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
25	11, 24	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
26	5, 6, 16, 18, 9, 10	homfeqval 17791	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
27	12, 26	eleqtrd 2864	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 17794	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
29	3, 13, 28	3eqtr3d 2805	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4593  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Hom chom 17359  compcco 17360  Hom_f chomf 17760  comp_fccomf 17761

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-homf 17764  df-comf 17765

This theorem is used by:  catpropd  17803  cidpropd  17804  oppccomfpropd  17821  monpropd  17832  funcpropd  17997  natpropd  18074  fucpropd  18075  xpcpropd  18302  hofpropd  18361  sectpropdlem  49970  uppropd  50115