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Theorem evlfcllem 17082
 Description: Lemma for evlfcl 17083. (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 2760 . . . 4 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2760 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
6 eqid 2760 . . . 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 477 . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
10 evlfcl.h . . . . 5 (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
1110simpld 477 . . . 4 (𝜑𝐻 ∈ (𝐶 Func 𝐷))
128simprd 482 . . . 4 (𝜑𝑋 ∈ (Base‘𝐶))
1310simprd 482 . . . 4 (𝜑𝑍 ∈ (Base‘𝐶))
14 eqid 2760 . . . 4 (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐻, 𝑍⟩)
15 evlfcl.q . . . . 5 𝑄 = (𝐶 FuncCat 𝐷)
16 eqid 2760 . . . . 5 (comp‘𝑄) = (comp‘𝑄)
17 evlfcl.a . . . . . 6 (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
1817simpld 477 . . . . 5 (𝜑𝐴 ∈ (𝐹𝑁𝐺))
19 evlfcl.b . . . . . 6 (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
2019simpld 477 . . . . 5 (𝜑𝐵 ∈ (𝐺𝑁𝐻))
2115, 7, 16, 18, 20fuccocl 16845 . . . 4 (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22 eqid 2760 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
23 evlfcl.g . . . . . 6 (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
2423simprd 482 . . . . 5 (𝜑𝑌 ∈ (Base‘𝐶))
2517simprd 482 . . . . 5 (𝜑𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))
2619simprd 482 . . . . 5 (𝜑𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))
274, 5, 22, 2, 12, 24, 13, 25, 26catcocl 16567 . . . 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 17080 . . 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 16844 . . . 4 (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍) = ((𝐵𝑍)(⟨((1st𝐹)‘𝑍), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))(𝐴𝑍)))
3029oveq1d 6829 . . 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 16743 . . . . . . 7 Rel (𝐶 Func 𝐷)
32 1st2ndbr 7385 . . . . . . 7 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3331, 9, 32sylancr 698 . . . . . 6 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
344, 5, 22, 6, 33, 12, 24, 13, 25, 26funcco 16752 . . . . 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 6830 . . . 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 16832 . . . . . . . . 9 (𝜑𝐴 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
377, 36, 4, 5, 6, 24, 13, 26nati 16836 . . . . . . . 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 6830 . . . . . . 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 2760 . . . . . . . 8 (Base‘𝐷) = (Base‘𝐷)
40 eqid 2760 . . . . . . . 8 (Hom ‘𝐷) = (Hom ‘𝐷)
414, 39, 33funcf1 16747 . . . . . . . . 9 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
4241, 24ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑌) ∈ (Base‘𝐷))
4341, 13ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((1st𝐹)‘𝑍) ∈ (Base‘𝐷))
4423simpld 477 . . . . . . . . . . 11 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
45 1st2ndbr 7385 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
4631, 44, 45sylancr 698 . . . . . . . . . 10 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
474, 39, 46funcf1 16747 . . . . . . . . 9 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
4847, 13ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑍) ∈ (Base‘𝐷))
494, 5, 40, 33, 24, 13funcf2 16749 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐹)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
5049, 26ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐹)𝑍)‘𝐿) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐹)‘𝑍)))
517, 36, 4, 40, 13natcl 16834 . . . . . . . 8 (𝜑 → (𝐴𝑍) ∈ (((1st𝐹)‘𝑍)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
52 1st2ndbr 7385 . . . . . . . . . . 11 ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
5331, 11, 52sylancr 698 . . . . . . . . . 10 (𝜑 → (1st𝐻)(𝐶 Func 𝐷)(2nd𝐻))
544, 39, 53funcf1 16747 . . . . . . . . 9 (𝜑 → (1st𝐻):(Base‘𝐶)⟶(Base‘𝐷))
5554, 13ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((1st𝐻)‘𝑍) ∈ (Base‘𝐷))
567, 20nat1st2nd 16832 . . . . . . . . 9 (𝜑𝐵 ∈ (⟨(1st𝐺), (2nd𝐺)⟩𝑁⟨(1st𝐻), (2nd𝐻)⟩))
577, 56, 4, 40, 13natcl 16834 . . . . . . . 8 (𝜑 → (𝐵𝑍) ∈ (((1st𝐺)‘𝑍)(Hom ‘𝐷)((1st𝐻)‘𝑍)))
5839, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57catass 16568 . . . . . . 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, 24ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((1st𝐺)‘𝑌) ∈ (Base‘𝐷))
607, 36, 4, 40, 24natcl 16834 . . . . . . . 8 (𝜑 → (𝐴𝑌) ∈ (((1st𝐹)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑌)))
614, 5, 40, 46, 24, 13funcf2 16749 . . . . . . . . 9 (𝜑 → (𝑌(2nd𝐺)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6261, 26ffvelrnd 6524 . . . . . . . 8 (𝜑 → ((𝑌(2nd𝐺)𝑍)‘𝐿) ∈ (((1st𝐺)‘𝑌)(Hom ‘𝐷)((1st𝐺)‘𝑍)))
6339, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57catass 16568 . . . . . . 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 2804 . . . . . 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 6829 . . . . 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, 12ffvelrnd 6524 . . . . . 6 (𝜑 → ((1st𝐹)‘𝑋) ∈ (Base‘𝐷))
674, 5, 40, 33, 12, 24funcf2 16749 . . . . . . 7 (𝜑 → (𝑋(2nd𝐹)𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶(((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6867, 25ffvelrnd 6524 . . . . . 6 (𝜑 → ((𝑋(2nd𝐹)𝑌)‘𝐾) ∈ (((1st𝐹)‘𝑋)(Hom ‘𝐷)((1st𝐹)‘𝑌)))
6939, 40, 6, 3, 43, 48, 55, 51, 57catcocl 16567 . . . . . 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 16568 . . . . 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 16567 . . . . . 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 16568 . . . . 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 2802 . . . 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 2794 . . 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 2798 . 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 2760 . . . . 5 (𝑄 ×c 𝐶) = (𝑄 ×c 𝐶)
7715fucbas 16841 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
7815, 7fuchom 16842 . . . . 5 𝑁 = (Hom ‘𝑄)
79 eqid 2760 . . . . 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 17048 . . . 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 6357 . . 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 6817 . . 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, 82syl6eqr 2812 . 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 6817 . . . . . 6 (𝐹(1st𝐸)𝑋) = ((1st𝐸)‘⟨𝐹, 𝑋⟩)
851, 2, 3, 4, 9, 12evlf1 17081 . . . . . 6 (𝜑 → (𝐹(1st𝐸)𝑋) = ((1st𝐹)‘𝑋))
8684, 85syl5eqr 2808 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐹, 𝑋⟩) = ((1st𝐹)‘𝑋))
87 df-ov 6817 . . . . . 6 (𝐺(1st𝐸)𝑌) = ((1st𝐸)‘⟨𝐺, 𝑌⟩)
881, 2, 3, 4, 44, 24evlf1 17081 . . . . . 6 (𝜑 → (𝐺(1st𝐸)𝑌) = ((1st𝐺)‘𝑌))
8987, 88syl5eqr 2808 . . . . 5 (𝜑 → ((1st𝐸)‘⟨𝐺, 𝑌⟩) = ((1st𝐺)‘𝑌))
9086, 89opeq12d 4561 . . . 4 (𝜑 → ⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩)
91 df-ov 6817 . . . . 5 (𝐻(1st𝐸)𝑍) = ((1st𝐸)‘⟨𝐻, 𝑍⟩)
921, 2, 3, 4, 11, 13evlf1 17081 . . . . 5 (𝜑 → (𝐻(1st𝐸)𝑍) = ((1st𝐻)‘𝑍))
9391, 92syl5eqr 2808 . . . 4 (𝜑 → ((1st𝐸)‘⟨𝐻, 𝑍⟩) = ((1st𝐻)‘𝑍))
9490, 93oveq12d 6832 . . 3 (𝜑 → (⟨((1st𝐸)‘⟨𝐹, 𝑋⟩), ((1st𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1st𝐸)‘⟨𝐻, 𝑍⟩)) = (⟨((1st𝐹)‘𝑋), ((1st𝐺)‘𝑌)⟩(comp‘𝐷)((1st𝐻)‘𝑍)))
95 df-ov 6817 . . . 4 (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96 eqid 2760 . . . . 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 17080 . . . 4 (𝜑 → (𝐵(⟨𝐺, 𝑌⟩(2nd𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((𝐵𝑍)(⟨((1st𝐺)‘𝑌), ((1st𝐺)‘𝑍)⟩(comp‘𝐷)((1st𝐻)‘𝑍))((𝑌(2nd𝐺)𝑍)‘𝐿)))
9895, 97syl5eqr 2808 . . 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 6817 . . . 4 (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100 eqid 2760 . . . . 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 17080 . . . 4 (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2nd𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((𝐴𝑌)(⟨((1st𝐹)‘𝑋), ((1st𝐹)‘𝑌)⟩(comp‘𝐷)((1st𝐺)‘𝑌))((𝑋(2nd𝐹)𝑌)‘𝐾)))
10299, 101syl5eqr 2808 . . 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 6835 . 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 2804 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 383   = wceq 1632   ∈ wcel 2139  ⟨cop 4327   class class class wbr 4804  Rel wrel 5271  ‘cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546   Func cfunc 16735   Nat cnat 16822   FuncCat cfuc 16823   ×c cxpc 17029   evalF cevlf 17070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-cat 16550  df-func 16739  df-nat 16824  df-fuc 16825  df-xpc 17033  df-evlf 17074 This theorem is referenced by:  evlfcl  17083
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