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Theorem comfeqval 16970
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 7165 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩(compf𝐶)𝑍) = (⟨𝑋, 𝑌⟩(compf𝐷)𝑍))
32oveqd 7165 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹))
4 eqid 2819 . . 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 16962 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐶)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹))
14 eqid 2819 . . 3 (compf𝐷) = (compf𝐷)
15 eqid 2819 . . 3 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2819 . . 3 (Hom ‘𝐷) = (Hom ‘𝐷)
17 comfeqval.2 . . 3 = (comp‘𝐷)
18 comfeqval.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
1918homfeqbas 16958 . . . . 5 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
205, 19syl5eq 2866 . . . 4 (𝜑𝐵 = (Base‘𝐷))
218, 20eleqtrd 2913 . . 3 (𝜑𝑋 ∈ (Base‘𝐷))
229, 20eleqtrd 2913 . . 3 (𝜑𝑌 ∈ (Base‘𝐷))
2310, 20eleqtrd 2913 . . 3 (𝜑𝑍 ∈ (Base‘𝐷))
245, 6, 16, 18, 8, 9homfeqval 16959 . . . 4 (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐷)𝑌))
2511, 24eleqtrd 2913 . . 3 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐷)𝑌))
265, 6, 16, 18, 9, 10homfeqval 16959 . . . 4 (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐷)𝑍))
2712, 26eleqtrd 2913 . . 3 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐷)𝑍))
2814, 15, 16, 17, 21, 22, 23, 25, 27comfval 16962 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(compf𝐷)𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
293, 13, 283eqtr3d 2862 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺(⟨𝑋, 𝑌 𝑍)𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cop 4565  cfv 6348  (class class class)co 7148  Basecbs 16475  Hom chom 16568  compcco 16569  Homf chomf 16929  compfccomf 16930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-homf 16933  df-comf 16934
This theorem is referenced by:  catpropd  16971  cidpropd  16972  oppccomfpropd  16989  monpropd  16999  funcpropd  17162  natpropd  17238  fucpropd  17239  xpcpropd  17450  hofpropd  17509
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