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Theorem comfeqval 16300
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 → (Homf𝐶) = (Homf𝐷))
comfeqval.4 (𝜑 → (compf𝐶) = (compf𝐷))
comfeqval.x (𝜑𝑋𝐵)
comfeqval.y (𝜑𝑌𝐵)
comfeqval.z (𝜑𝑍𝐵)
comfeqval.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
comfeqval.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
Assertion
Ref Expression
comfeqval (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))

Proof of Theorem comfeqval
StepHypRef Expression
1 comfeqval.4 . . . 4 (𝜑 → (compf𝐶) = (compf𝐷))
21oveqd 6627 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩(compf𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf𝐷)𝑍))
32oveqd 6627 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹))
4 eqid 2621 . . 3 (compf𝐶) = (compf𝐶)
5 comfeqval.b . . 3 𝐵 = (Base‘𝐶)
6 comfeqval.h . . 3 𝐻 = (Hom ‘𝐶)
7 comfeqval.1 . . 3 · = (comp‘𝐶)
8 comfeqval.x . . 3 (𝜑𝑋𝐵)
9 comfeqval.y . . 3 (𝜑𝑌𝐵)
10 comfeqval.z . . 3 (𝜑𝑍𝐵)
11 comfeqval.f . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
12 comfeqval.g . . 3 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
134, 5, 6, 7, 8, 9, 10, 11, 12comfval 16292 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
14 eqid 2621 . . 3 (compf𝐷) = (compf𝐷)
15 eqid 2621 . . 3 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2621 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
17 comfeqval.2 . . 3 = (comp‘𝐷)
18 comfeqval.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
1918homfeqbas 16288 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
205, 19syl5eq 2667 . . . 4 (𝜑𝐵 = (Base‘𝐷))
218, 20eleqtrd 2700 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
229, 20eleqtrd 2700 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
2310, 20eleqtrd 2700 . . 3 (𝜑𝑍 ∈ (Base‘𝐷))
245, 6, 16, 18, 8, 9homfeqval 16289 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
2511, 24eleqtrd 2700 . . 3 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
265, 6, 16, 18, 9, 10homfeqval 16289 . . . 4 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
2712, 26eleqtrd 2700 . . 3 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 16292 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
293, 13, 283eqtr3d 2663 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  cop 4159  cfv 5852  (class class class)co 6610  Basecbs 15792  Hom chom 15884  compcco 15885  Homf chomf 16259  compfccomf 16260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-homf 16263  df-comf 16264
This theorem is referenced by:  catpropd  16301  cidpropd  16302  oppccomfpropd  16319  monpropd  16329  funcpropd  16492  natpropd  16568  fucpropd  16569  xpcpropd  16780  hofpropd  16839
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