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Theorem comfeqval 17753
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 7448 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩(compf𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf𝐷)𝑍))
32oveqd 7448 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹))
4 eqid 2735 . . 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 17745 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
14 eqid 2735 . . 3 (compf𝐷) = (compf𝐷)
15 eqid 2735 . . 3 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2735 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
17 comfeqval.2 . . 3 = (comp‘𝐷)
18 comfeqval.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
1918homfeqbas 17741 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
205, 19eqtrid 2787 . . . 4 (𝜑𝐵 = (Base‘𝐷))
218, 20eleqtrd 2841 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
229, 20eleqtrd 2841 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
2310, 20eleqtrd 2841 . . 3 (𝜑𝑍 ∈ (Base‘𝐷))
245, 6, 16, 18, 8, 9homfeqval 17742 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
2511, 24eleqtrd 2841 . . 3 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
265, 6, 16, 18, 9, 10homfeqval 17742 . . . 4 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
2712, 26eleqtrd 2841 . . 3 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 17745 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
293, 13, 283eqtr3d 2783 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cop 4637  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  compcco 17310  Homf chomf 17711  compfccomf 17712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-homf 17715  df-comf 17716
This theorem is referenced by:  catpropd  17754  cidpropd  17755  oppccomfpropd  17774  monpropd  17785  funcpropd  17954  natpropd  18033  fucpropd  18034  xpcpropd  18265  hofpropd  18324
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