Theorem comfeqval 17334

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 7272	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍))
3	2	oveqd 7272	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹))
4		eqid 2738	. . 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 17326	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹))
14		eqid 2738	. . 3 ⊢ (compf‘𝐷) = (compf‘𝐷)
15		eqid 2738	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
16		eqid 2738	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
17		comfeqval.2	. . 3 ⊢ ∙ = (comp‘𝐷)
18		comfeqval.3	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
19	18	homfeqbas 17322	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
20	5, 19	eqtrid 2790	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐷))
21	8, 20	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐷))
22	9, 20	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐷))
23	10, 20	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐷))
24	5, 6, 16, 18, 8, 9	homfeqval 17323	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
25	11, 24	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
26	5, 6, 16, 18, 9, 10	homfeqval 17323	. . . 4 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
27	12, 26	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
28	14, 15, 16, 17, 21, 22, 23, 25, 27	comfval 17326	. 2 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf‘𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))
29	3, 13, 28	3eqtr3d 2786	1 ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩ ∙ 𝑍)𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Hom_f chomf 17292  comp_fccomf 17293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-homf 17296  df-comf 17297

This theorem is referenced by:  catpropd  17335  cidpropd  17336  oppccomfpropd  17355  monpropd  17366  funcpropd  17532  natpropd  17610  fucpropd  17611  xpcpropd  17842  hofpropd  17901