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Theorem evlfcllem 18301

Description: Lemma for evlfcl 18302. (Contributed by Mario Carneiro, 12-Jan-2017.)

**Hypotheses**
Ref	Expression
evlfcl.e	⊢ 𝐸 = (𝐶 eval_F 𝐷)
evlfcl.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
evlfcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
evlfcl.d	⊢ (𝜑 → 𝐷 ∈ 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⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×_c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))

**Proof of Theorem evlfcllem**
Step	Hyp	Ref	Expression
1		evlfcl.e	. . . 4 ⊢ 𝐸 = (𝐶 eval_F 𝐷)
2		evlfcl.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		evlfcl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
4		eqid 2765	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2765	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2765	. . . 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)))
9	8	simpld 500	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		evlfcl.h	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
11	10	simpld 500	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
12	8	simprd 501	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
13	10	simprd 501	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
14		eqid 2765	. . . 4 ⊢ (⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)
15		evlfcl.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
16		eqid 2765	. . . . 5 ⊢ (comp‘𝑄) = (comp‘𝑄)
17		evlfcl.a	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
18	17	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
19		evlfcl.b	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
20	19	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐺𝑁𝐻))
21	15, 7, 16, 18, 20	fuccocl 18048	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22		eqid 2765	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
23		evlfcl.g	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
24	23	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
25	17	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))
26	19	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))
27	4, 5, 22, 2, 12, 24, 13, 25, 26	catcocl 17765	. . . 4 ⊢ (𝜑 → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾) ∈ (𝑋(Hom ‘𝐶)𝑍))
28	1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 21, 27	evlf2val 18299	. . 3 ⊢ (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))))
29	15, 7, 4, 6, 16, 18, 20, 13	fuccoval 18047	. . . 4 ⊢ (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍) = ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍)))
30	29	oveq1d 7434	. . 3 ⊢ (𝜑 → (((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)‘𝑍)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))))
31		relfunc 17943	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
32		1st2ndbr 8045	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
33	31, 9, 32	sylancr 599	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
34	4, 5, 22, 6, 33, 12, 24, 13, 25, 26	funcco 17952	. . . . 5 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)))
35	34	oveq2d 7435	. . . 4 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
36	7, 18	nat1st2nd 18035	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
37	7, 36, 4, 5, 6, 24, 13, 26	nati 18039	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)) = (((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))(𝐴‘𝑌)))
38	37	oveq2d 7435	. . . . . . 7 ⊢ (𝜑 → ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿))) = ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))(𝐴‘𝑌))))
39		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
40		eqid 2765	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
41	4, 39, 33	funcf1 17947	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
42	41, 24	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ (Base‘𝐷))
43	41, 13	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ (Base‘𝐷))
44	23	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
45		1st2ndbr 8045	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
46	31, 44, 45	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
47	4, 39, 46	funcf1 17947	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
48	47, 13	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑍) ∈ (Base‘𝐷))
49	4, 5, 40, 33, 24, 13	funcf2 17949	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(2^nd ‘𝐹)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑍)))
50	49, 26	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐹)𝑍)‘𝐿) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑍)))
51	7, 36, 4, 40, 13	natcl 18037	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ∈ (((1^st ‘𝐹)‘𝑍)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
52		1st2ndbr 8045	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
53	31, 11, 52	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
54	4, 39, 53	funcf1 17947	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
55	54, 13	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐻)‘𝑍) ∈ (Base‘𝐷))
56	7, 20	nat1st2nd 18035	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
57	7, 56, 4, 40, 13	natcl 18037	. . . . . . . 8 ⊢ (𝜑 → (𝐵‘𝑍) ∈ (((1^st ‘𝐺)‘𝑍)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑍)))
58	39, 40, 6, 3, 42, 43, 48, 50, 51, 55, 57	catass 17766	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)) = ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿))))
59	47, 24	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑌) ∈ (Base‘𝐷))
60	7, 36, 4, 40, 24	natcl 18037	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑌)))
61	4, 5, 40, 46, 24, 13	funcf2 17949	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(2^nd ‘𝐺)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1^st ‘𝐺)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
62	61, 26	ffvelcdmd 7084	. . . . . . . 8 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐺)𝑍)‘𝐿) ∈ (((1^st ‘𝐺)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
63	39, 40, 6, 3, 42, 59, 48, 60, 62, 55, 57	catass 17766	. . . . . . 7 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑌)) = ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑍))(𝐴‘𝑌))))
64	38, 58, 63	3eqtr4d 2810	. . . . . 6 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑌)))
65	64	oveq1d 7434	. . . . 5 ⊢ (𝜑 → ((((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)) = ((((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑌))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)))
66	41, 12	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ (Base‘𝐷))
67	4, 5, 40, 33, 12, 24	funcf2 17949	. . . . . . 7 ⊢ (𝜑 → (𝑋(2^nd ‘𝐹)𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑌)))
68	67, 25	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → ((𝑋(2^nd ‘𝐹)𝑌)‘𝐾) ∈ (((1^st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑌)))
69	39, 40, 6, 3, 43, 48, 55, 51, 57	catcocl 17765	. . . . . 6 ⊢ (𝜑 → ((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍)) ∈ (((1^st ‘𝐹)‘𝑍)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑍)))
70	39, 40, 6, 3, 66, 42, 43, 68, 50, 55, 69	catass 17766	. . . . 5 ⊢ (𝜑 → ((((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)) = (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
71	39, 40, 6, 3, 59, 48, 55, 62, 57	catcocl 17765	. . . . . 6 ⊢ (𝜑 → ((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)) ∈ (((1^st ‘𝐺)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑍)))
72	39, 40, 6, 3, 66, 42, 59, 68, 60, 55, 71	catass 17766	. . . . 5 ⊢ (𝜑 → ((((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑌))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
73	65, 70, 72	3eqtr3d 2808	. . . 4 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(((𝑌(2^nd ‘𝐹)𝑍)‘𝐿)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐹)‘𝑍))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
74	35, 73	eqtrd 2800	. . 3 ⊢ (𝜑 → (((𝐵‘𝑍)(⟨((1^st ‘𝐹)‘𝑍), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))(𝐴‘𝑍))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑋(2^nd ‘𝐹)𝑍)‘(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾))) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
75	28, 30, 74	3eqtrd 2804	. 2 ⊢ (𝜑 → ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
76		eqid 2765	. . . . 5 ⊢ (𝑄 ×_c 𝐶) = (𝑄 ×_c 𝐶)
77	15	fucbas 18044	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
78	15, 7	fuchom 18045	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
79		eqid 2765	. . . . 5 ⊢ (comp‘(𝑄 ×_c 𝐶)) = (comp‘(𝑄 ×_c 𝐶))
80	76, 77, 4, 78, 5, 9, 12, 44, 24, 16, 22, 79, 11, 13, 18, 25, 20, 26	xpcco2 18267	. . . 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)⟩)
81	80	fveq2d 6889	. . 3 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×_c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨(𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴), (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)⟩))
82		df-ov 7422	. . 3 ⊢ ((𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴)(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)) = ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨(𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴), (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)⟩)
83	81, 82	eqtr4di 2818	. 2 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨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⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝐾)))
84		df-ov 7422	. . . . . 6 ⊢ (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩)
85	1, 2, 3, 4, 9, 12	evlf1 18300	. . . . . 6 ⊢ (𝜑 → (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐹)‘𝑋))
86	84, 85	eqtr3id 2814	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩) = ((1^st ‘𝐹)‘𝑋))
87		df-ov 7422	. . . . . 6 ⊢ (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)
88	1, 2, 3, 4, 44, 24	evlf1 18300	. . . . . 6 ⊢ (𝜑 → (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐺)‘𝑌))
89	87, 88	eqtr3id 2814	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩) = ((1^st ‘𝐺)‘𝑌))
90	86, 89	opeq12d 4848	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩)
91		df-ov 7422	. . . . 5 ⊢ (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩)
92	1, 2, 3, 4, 11, 13	evlf1 18300	. . . . 5 ⊢ (𝜑 → (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐻)‘𝑍))
93	91, 92	eqtr3id 2814	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩) = ((1^st ‘𝐻)‘𝑍))
94	90, 93	oveq12d 7437	. . 3 ⊢ (𝜑 → (⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩)) = (⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍)))
95		df-ov 7422	. . . 4 ⊢ (𝐵(⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96		eqid 2765	. . . . 5 ⊢ (⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩) = (⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)
97	1, 2, 3, 4, 5, 6, 7, 44, 11, 24, 13, 96, 20, 26	evlf2val 18299	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)))
98	95, 97	eqtr3id 2814	. . 3 ⊢ (𝜑 → ((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩) = ((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿)))
99		df-ov 7422	. . . 4 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100		eqid 2765	. . . . 5 ⊢ (⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩) = (⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)
101	1, 2, 3, 4, 5, 6, 7, 9, 44, 12, 24, 100, 18, 25	evlf2val 18299	. . . 4 ⊢ (𝜑 → (𝐴(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)))
102	99, 101	eqtr3id 2814	. . 3 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩) = ((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾)))
103	94, 98, 102	oveq123d 7440	. 2 ⊢ (𝜑 → (((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)) = (((𝐵‘𝑍)(⟨((1^st ‘𝐺)‘𝑌), ((1^st ‘𝐺)‘𝑍)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝑌(2^nd ‘𝐺)𝑍)‘𝐿))(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑍))((𝐴‘𝑌)(⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐹)‘𝑌)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑌))((𝑋(2^nd ‘𝐹)𝑌)‘𝐾))))
104	75, 83, 103	3eqtr4d 2810	1 ⊢ (𝜑 → ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘(⟨𝐵, 𝐿⟩(⟨⟨𝐹, 𝑋⟩, ⟨𝐺, 𝑌⟩⟩(comp‘(𝑄 ×_c 𝐶))⟨𝐻, 𝑍⟩)⟨𝐴, 𝐾⟩)) = (((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)(⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩(comp‘𝐷)((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩))((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⟨cop 4597 class class class wbr 5111 Rel wrel 5668 ‘cfv 6540 (class class class)co 7419 1^st c1st 7990 2^nd c2nd 7991 Basecbs 17293 Hom chom 17345 compcco 17346 Catccat 17744 Func cfunc 17935 Nat cnat 18025 FuncCat cfuc 18026 ×_c cxpc 18248 eval_F cevlf 18289

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17231 df-slot 17266 df-ndx 17278 df-base 17294 df-hom 17358 df-cco 17359 df-cat 17748 df-func 17939 df-nat 18027 df-fuc 18028 df-xpc 18252 df-evlf 18293

This theorem is used by: evlfcl 18302

W3C validator