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Theorem evlfcllem 18246
Description: Lemma for evlfcl 18247. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfcl.e 𝐸 = (𝐶 evalF 𝐷)
evlfcl.q 𝑄 = (𝐶 FuncCat 𝐷)
evlfcl.c (𝜑𝐶 ∈ Cat)
evlfcl.d (𝜑𝐷 ∈ Cat)
evlfcl.n 𝑁 = (𝐶 Nat 𝐷)
evlfcl.f (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (Base‘𝐶)))
evlfcl.g (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
evlfcl.h (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
evlfcl.a (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
evlfcl.b (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
Assertion
Ref Expression
evlfcllem (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))

Proof of Theorem evlfcllem
StepHypRef Expression
1 evlfcl.e . . . 4 𝐸 = (𝐶 evalF 𝐷)
2 evlfcl.c . . . 4 (𝜑𝐶 ∈ Cat)
3 evlfcl.d . . . 4 (𝜑𝐷 ∈ Cat)
4 eqid 2726 . . . 4 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2726 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2726 . . . 4 (comp‘𝐷) = (comp‘𝐷)
7 evlfcl.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
8 evlfcl.f . . . . 5 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑋 ∈ (Base‘𝐶)))
98simpld 493 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
10 evlfcl.h . . . . 5 (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
1110simpld 493 . . . 4 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
128simprd 494 . . . 4 (𝜑𝑋 ∈ (Base‘𝐶))
1310simprd 494 . . . 4 (𝜑𝑍 ∈ (Base‘𝐶))
14 eqid 2726 . . . 4 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)
15 evlfcl.q . . . . 5 𝑄 = (𝐶 FuncCat 𝐷)
16 eqid 2726 . . . . 5 (comp‘𝑄) = (comp‘𝑄)
17 evlfcl.a . . . . . 6 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
1817simpld 493 . . . . 5 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
19 evlfcl.b . . . . . 6 (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
2019simpld 493 . . . . 5 (𝜑𝐵 ∈ (𝐺𝑁𝐻))
2115, 7, 16, 18, 20fuccocl 17989 . . . 4 (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22 eqid 2726 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
23 evlfcl.g . . . . . 6 (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
2423simprd 494 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2517simprd 494 . . . . 5 (𝜑𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))
2619simprd 494 . . . . 5 (𝜑𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 17698 . . . 4 (𝜑 → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾) ∈ (𝑋(Hom ‘𝐶)𝑍))
281, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 21, 27evlf2val 18244 . . 3 (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))))
2915, 7, 4, 6, 16, 18, 20, 13fuccoval 17988 . . . 4 (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍)))
3029oveq1d 7439 . . 3 (𝜑 → (((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))))
31 relfunc 17881 . . . . . . 7 Rel (𝐶 Func 𝐷)
32 1st2ndbr 8056 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3331, 9, 32sylancr 585 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 17890 . . . . 5 (𝜑 → ((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝑌(2nd𝐹)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐹)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾)))
3534oveq2d 7440 . . . 4 (𝜑 → (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(((𝑌(2nd𝐹)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐹)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾))))
367, 18nat1st2nd 17974 . . . . . . . . 9 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
377, 36, 4, 5, 6, 24, 13, 26nati 17978 . . . . . . . 8 (𝜑 → ((𝐴𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐺)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿)) = (((𝑌(2nd𝐺)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑍))(𝐴𝑌)))
3837oveq2d 7440 . . . . . . 7 (𝜑 → ((𝐵𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐺)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿))) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(((𝑌(2nd𝐺)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑍))(𝐴𝑌))))
39 eqid 2726 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
40 eqid 2726 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
414, 39, 33funcf1 17885 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
4241, 24ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑌) ∈ (Base‘𝐷))
4341, 13ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑍) ∈ (Base‘𝐷))
4423simpld 493 . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
45 1st2ndbr 8056 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4631, 44, 45sylancr 585 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
474, 39, 46funcf1 17885 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
4847, 13ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑍) ∈ (Base‘𝐷))
494, 5, 40, 33, 24, 13funcf2 17887 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐹)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
5049, 26ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐹)𝑍)‘𝐿) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
517, 36, 4, 40, 13natcl 17976 . . . . . . . 8 (𝜑 → (𝐴𝑍) ∈ (((1st𝐹)‘𝑍)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
52 1st2ndbr 8056 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
5331, 11, 52sylancr 585 . . . . . . . . . 10 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
544, 39, 53funcf1 17885 . . . . . . . . 9 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
5554, 13ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((1st𝐻)‘𝑍) ∈ (Base‘𝐷))
567, 20nat1st2nd 17974 . . . . . . . . 9 (𝜑𝐵 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
577, 56, 4, 40, 13natcl 17976 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ (((1st𝐺)‘𝑍)(Hom ‘𝐷)((1st𝐻)‘𝑍)))
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 17699 . . . . . . 7 (𝜑 → (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿)) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐺)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿))))
5947, 24ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑌) ∈ (Base‘𝐷))
607, 36, 4, 40, 24natcl 17976 . . . . . . . 8 (𝜑 → (𝐴𝑌) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑌)))
614, 5, 40, 46, 24, 13funcf2 17887 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐺)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6261, 26ffvelcdmd 7099 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐺)𝑍)‘𝐿) ∈ (((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6339, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57catass 17699 . . . . . . 7 (𝜑 → (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑌)) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(((𝑌(2nd𝐺)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑍))(𝐴𝑌))))
6438, 58, 633eqtr4d 2776 . . . . . 6 (𝜑 → (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿)) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑌)))
6564oveq1d 7439 . . . . 5 (𝜑 → ((((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾)) = ((((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑌))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾)))
6641, 12ffvelcdmd 7099 . . . . . 6 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝐷))
674, 5, 40, 33, 12, 24funcf2 17887 . . . . . . 7 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6867, 25ffvelcdmd 7099 . . . . . 6 (𝜑 → ((𝑋(2nd𝐹)𝑌)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6939, 40, 6, 3, 43, 48, 55, 51, 57catcocl 17698 . . . . . 6 (𝜑 → ((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍)) ∈ (((1st𝐹)‘𝑍)(Hom ‘𝐷)((1st𝐻)‘𝑍)))
7039, 40, 6, 3, 66, 42, 43, 68, 50, 55, 69catass 17699 . . . . 5 (𝜑 → ((((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑌), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐹)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾)) = (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(((𝑌(2nd𝐹)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐹)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾))))
7139, 40, 6, 3, 59, 48, 55, 62, 57catcocl 17698 . . . . . 6 (𝜑 → ((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿)) ∈ (((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐻)‘𝑍)))
7239, 40, 6, 3, 66, 42, 59, 68, 60, 55, 71catass 17699 . . . . 5 (𝜑 → ((((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑌), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑌))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾)) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾))))
7365, 70, 723eqtr3d 2774 . . . 4 (𝜑 → (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(((𝑌(2nd𝐹)𝑍)‘𝐿)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐹)‘𝑍))((𝑋(2nd𝐹)𝑌)‘𝐾))) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾))))
7435, 73eqtrd 2766 . . 3 (𝜑 → (((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍))(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑋(2nd𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾))))
7528, 30, 743eqtrd 2770 . 2 (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾))))
76 eqid 2726 . . . . 5 (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
7715fucbas 17984 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
7815, 7fuchom 17985 . . . . 5 𝑁 = (Hom ‘𝑄)
79 eqid 2726 . . . . 5 (comp‘(𝑄 ×c 𝐶)) = (comp‘(𝑄 ×c 𝐶))
8076, 77, 4, 78, 5, 9, 12, 44, 24, 16, 22, 79, 11, 13, 18, 25, 20, 26xpcco2 18211 . . . 4 (𝜑 → (⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩) = ⟨(𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴), (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)⟩)
8180fveq2d 6905 . . 3 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨(𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴), (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)⟩))
82 df-ov 7427 . . 3 ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨(𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴), (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)⟩)
8381, 82eqtr4di 2784 . 2 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)))
84 df-ov 7427 . . . . . 6 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
851, 2, 3, 4, 9, 12evlf1 18245 . . . . . 6 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
8684, 85eqtr3id 2780 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐹, 𝑋⟩) = ((1st𝐹)‘𝑋))
87 df-ov 7427 . . . . . 6 (𝐺(1st𝐸)𝑌) = ((1st𝐸)‘⟨𝐺, 𝑌⟩)
881, 2, 3, 4, 44, 24evlf1 18245 . . . . . 6 (𝜑 → (𝐺(1st𝐸)𝑌) = ((1st𝐺)‘𝑌))
8987, 88eqtr3id 2780 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐺, 𝑌⟩) = ((1st𝐺)‘𝑌))
9086, 89opeq12d 4887 . . . 4 (𝜑 → ⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩)
91 df-ov 7427 . . . . 5 (𝐻(1st𝐸)𝑍) = ((1st𝐸)‘⟨𝐻, 𝑍⟩)
921, 2, 3, 4, 11, 13evlf1 18245 . . . . 5 (𝜑 → (𝐻(1st𝐸)𝑍) = ((1st𝐻)‘𝑍))
9391, 92eqtr3id 2780 . . . 4 (𝜑 → ((1st𝐸)‘⟨𝐻, 𝑍⟩) = ((1st𝐻)‘𝑍))
9490, 93oveq12d 7442 . . 3 (𝜑 → (⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩)) = (⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍)))
95 df-ov 7427 . . . 4 (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96 eqid 2726 . . . . 5 (⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)
971, 2, 3, 4, 5, 6, 7, 44, 11, 24, 13, 96, 20, 26evlf2val 18244 . . . 4 (𝜑 → (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿)))
9895, 97eqtr3id 2780 . . 3 (𝜑 → ((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩) = ((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿)))
99 df-ov 7427 . . . 4 (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100 eqid 2726 . . . . 5 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)
1011, 2, 3, 4, 5, 6, 7, 9, 44, 12, 24, 100, 18, 25evlf2val 18244 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
10299, 101eqtr3id 2780 . . 3 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
10394, 98, 102oveq123d 7445 . 2 (𝜑 → (((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)) = (((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿))(⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾))))
10475, 83, 1033eqtr4d 2776 1 (𝜑 → ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cop 4639   class class class wbr 5153  Rel wrel 5687  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  Basecbs 17213  Hom chom 17277  compcco 17278  Catccat 17677   Func cfunc 17873   Nat cnat 17964   FuncCat cfuc 17965   ×c cxpc 18192   evalF cevlf 18234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-slot 17184  df-ndx 17196  df-base 17214  df-hom 17290  df-cco 17291  df-cat 17681  df-func 17877  df-nat 17966  df-fuc 17967  df-xpc 18196  df-evlf 18238
This theorem is referenced by:  evlfcl  18247
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