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Theorem evlfcllem 18212
Description: Lemma for evlfcl 18213. (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 2725 . . . 4 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2725 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2725 . . . 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 2725 . . . 4 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)
15 evlfcl.q . . . . 5 𝑄 = (𝐶 FuncCat 𝐷)
16 eqid 2725 . . . . 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 17955 . . . 4 (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22 eqid 2725 . . . . 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 17664 . . . 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 18210 . . 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 17954 . . . 4 (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍)))
3029oveq1d 7431 . . 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 17847 . . . . . . 7 Rel (𝐶 Func 𝐷)
32 1st2ndbr 8044 . . . . . . 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 17856 . . . . 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 7432 . . . 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 17940 . . . . . . . . 9 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
377, 36, 4, 5, 6, 24, 13, 26nati 17944 . . . . . . . 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 7432 . . . . . . 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 2725 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
40 eqid 2725 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
414, 39, 33funcf1 17851 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
4241, 24ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑌) ∈ (Base‘𝐷))
4341, 13ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑍) ∈ (Base‘𝐷))
4423simpld 493 . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
45 1st2ndbr 8044 . . . . . . . . . . 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 17851 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
4847, 13ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑍) ∈ (Base‘𝐷))
494, 5, 40, 33, 24, 13funcf2 17853 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐹)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
5049, 26ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐹)𝑍)‘𝐿) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
517, 36, 4, 40, 13natcl 17942 . . . . . . . 8 (𝜑 → (𝐴𝑍) ∈ (((1st𝐹)‘𝑍)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
52 1st2ndbr 8044 . . . . . . . . . . 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 17851 . . . . . . . . 9 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
5554, 13ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((1st𝐻)‘𝑍) ∈ (Base‘𝐷))
567, 20nat1st2nd 17940 . . . . . . . . 9 (𝜑𝐵 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
577, 56, 4, 40, 13natcl 17942 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ (((1st𝐺)‘𝑍)(Hom ‘𝐷)((1st𝐻)‘𝑍)))
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 17665 . . . . . . 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 7090 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑌) ∈ (Base‘𝐷))
607, 36, 4, 40, 24natcl 17942 . . . . . . . 8 (𝜑 → (𝐴𝑌) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑌)))
614, 5, 40, 46, 24, 13funcf2 17853 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐺)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6261, 26ffvelcdmd 7090 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐺)𝑍)‘𝐿) ∈ (((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6339, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57catass 17665 . . . . . . 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 2775 . . . . . 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 7431 . . . . 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 7090 . . . . . 6 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝐷))
674, 5, 40, 33, 12, 24funcf2 17853 . . . . . . 7 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6867, 25ffvelcdmd 7090 . . . . . 6 (𝜑 → ((𝑋(2nd𝐹)𝑌)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6939, 40, 6, 3, 43, 48, 55, 51, 57catcocl 17664 . . . . . 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 17665 . . . . 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 17664 . . . . . 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 17665 . . . . 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 2773 . . . 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 2765 . . 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 2769 . 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 2725 . . . . 5 (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
7715fucbas 17950 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
7815, 7fuchom 17951 . . . . 5 𝑁 = (Hom ‘𝑄)
79 eqid 2725 . . . . 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 18177 . . . 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 6896 . . 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 7419 . . 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 2783 . 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 7419 . . . . . 6 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
851, 2, 3, 4, 9, 12evlf1 18211 . . . . . 6 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
8684, 85eqtr3id 2779 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐹, 𝑋⟩) = ((1st𝐹)‘𝑋))
87 df-ov 7419 . . . . . 6 (𝐺(1st𝐸)𝑌) = ((1st𝐸)‘⟨𝐺, 𝑌⟩)
881, 2, 3, 4, 44, 24evlf1 18211 . . . . . 6 (𝜑 → (𝐺(1st𝐸)𝑌) = ((1st𝐺)‘𝑌))
8987, 88eqtr3id 2779 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐺, 𝑌⟩) = ((1st𝐺)‘𝑌))
9086, 89opeq12d 4877 . . . 4 (𝜑 → ⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩)
91 df-ov 7419 . . . . 5 (𝐻(1st𝐸)𝑍) = ((1st𝐸)‘⟨𝐻, 𝑍⟩)
921, 2, 3, 4, 11, 13evlf1 18211 . . . . 5 (𝜑 → (𝐻(1st𝐸)𝑍) = ((1st𝐻)‘𝑍))
9391, 92eqtr3id 2779 . . . 4 (𝜑 → ((1st𝐸)‘⟨𝐻, 𝑍⟩) = ((1st𝐻)‘𝑍))
9490, 93oveq12d 7434 . . 3 (𝜑 → (⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩)) = (⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍)))
95 df-ov 7419 . . . 4 (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96 eqid 2725 . . . . 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 18210 . . . 4 (𝜑 → (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿)))
9895, 97eqtr3id 2779 . . 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 7419 . . . 4 (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100 eqid 2725 . . . . 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 18210 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
10299, 101eqtr3id 2779 . . 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 7437 . 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 2775 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 1533  wcel 2098  cop 4630   class class class wbr 5143  Rel wrel 5677  cfv 6543  (class class class)co 7416  1st c1st 7989  2nd c2nd 7990  Basecbs 17179  Hom chom 17243  compcco 17244  Catccat 17643   Func cfunc 17839   Nat cnat 17930   FuncCat cfuc 17931   ×c cxpc 18158   evalF cevlf 18200
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-hom 17256  df-cco 17257  df-cat 17647  df-func 17843  df-nat 17932  df-fuc 17933  df-xpc 18162  df-evlf 18204
This theorem is referenced by:  evlfcl  18213
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