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Theorem evlfcllem 18127

Description: Lemma for evlfcl 18128. (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 2731	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2731	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2731	. . . 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 494	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		evlfcl.h	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
11	10	simpld 494	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
12	8	simprd 495	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
13	10	simprd 495	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
14		eqid 2731	. . . 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 2731	. . . . 5 ⊢ (comp‘𝑄) = (comp‘𝑄)
17		evlfcl.a	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
18	17	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
19		evlfcl.b	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
20	19	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐺𝑁𝐻))
21	15, 7, 16, 18, 20	fuccocl 17874	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22		eqid 2731	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
23		evlfcl.g	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
24	23	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
25	17	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))
26	19	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))
27	4, 5, 22, 2, 12, 24, 13, 25, 26	catcocl 17591	. . . 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 18125	. . 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 17873	. . . 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 7361	. . 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 17769	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
32		1st2ndbr 7974	. . . . . . 7 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
33	31, 9, 32	sylancr 587	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
34	4, 5, 22, 6, 33, 12, 24, 13, 25, 26	funcco 17778	. . . . 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 7362	. . . 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 17861	. . . . . . . . 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 17865	. . . . . . . 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 7362	. . . . . . 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 2731	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
40		eqid 2731	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
41	4, 39, 33	funcf1 17773	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
42	41, 24	ffvelcdmd 7018	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ (Base‘𝐷))
43	41, 13	ffvelcdmd 7018	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ (Base‘𝐷))
44	23	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
45		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
46	31, 44, 45	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
47	4, 39, 46	funcf1 17773	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
48	47, 13	ffvelcdmd 7018	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑍) ∈ (Base‘𝐷))
49	4, 5, 40, 33, 24, 13	funcf2 17775	. . . . . . . . 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 7018	. . . . . . . 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 17863	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ∈ (((1^st ‘𝐹)‘𝑍)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
52		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
53	31, 11, 52	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
54	4, 39, 53	funcf1 17773	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
55	54, 13	ffvelcdmd 7018	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐻)‘𝑍) ∈ (Base‘𝐷))
56	7, 20	nat1st2nd 17861	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
57	7, 56, 4, 40, 13	natcl 17863	. . . . . . . 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 17592	. . . . . . 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 7018	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑌) ∈ (Base‘𝐷))
60	7, 36, 4, 40, 24	natcl 17863	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑌)))
61	4, 5, 40, 46, 24, 13	funcf2 17775	. . . . . . . . 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 7018	. . . . . . . 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 17592	. . . . . . 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 2776	. . . . . 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 7361	. . . . 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 7018	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ (Base‘𝐷))
67	4, 5, 40, 33, 12, 24	funcf2 17775	. . . . . . 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 7018	. . . . . 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 17591	. . . . . 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 17592	. . . . 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 17591	. . . . . 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 17592	. . . . 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 2774	. . . 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 2766	. . 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 2770	. 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 2731	. . . . 5 ⊢ (𝑄 ×_c 𝐶) = (𝑄 ×_c 𝐶)
77	15	fucbas 17870	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
78	15, 7	fuchom 17871	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
79		eqid 2731	. . . . 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 18093	. . . 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 6826	. . 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 7349	. . 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 2784	. 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 7349	. . . . . 6 ⊢ (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩)
85	1, 2, 3, 4, 9, 12	evlf1 18126	. . . . . 6 ⊢ (𝜑 → (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐹)‘𝑋))
86	84, 85	eqtr3id 2780	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩) = ((1^st ‘𝐹)‘𝑋))
87		df-ov 7349	. . . . . 6 ⊢ (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)
88	1, 2, 3, 4, 44, 24	evlf1 18126	. . . . . 6 ⊢ (𝜑 → (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐺)‘𝑌))
89	87, 88	eqtr3id 2780	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩) = ((1^st ‘𝐺)‘𝑌))
90	86, 89	opeq12d 4830	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩)
91		df-ov 7349	. . . . 5 ⊢ (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩)
92	1, 2, 3, 4, 11, 13	evlf1 18126	. . . . 5 ⊢ (𝜑 → (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐻)‘𝑍))
93	91, 92	eqtr3id 2780	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩) = ((1^st ‘𝐻)‘𝑍))
94	90, 93	oveq12d 7364	. . 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 7349	. . . 4 ⊢ (𝐵(⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96		eqid 2731	. . . . 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 18125	. . . 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 2780	. . 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 7349	. . . 4 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100		eqid 2731	. . . . 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 18125	. . . 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 2780	. . 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 7367	. 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 2776	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

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4579 class class class wbr 5089 Rel wrel 5619 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 Basecbs 17120 Hom chom 17172 compcco 17173 Catccat 17570 Func cfunc 17761 Nat cnat 17851 FuncCat cfuc 17852 ×_c cxpc 18074 eval_F cevlf 18115

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-func 17765 df-nat 17853 df-fuc 17854 df-xpc 18078 df-evlf 18119

This theorem is referenced by: evlfcl 18128

W3C validator