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Theorem evlfcllem 17939

Description: Lemma for evlfcl 17940. (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 2738	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2738	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2738	. . . 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 495	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		evlfcl.h	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝑍 ∈ (Base‘𝐶)))
11	10	simpld 495	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
12	8	simprd 496	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
13	10	simprd 496	. . . 4 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
14		eqid 2738	. . . 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 2738	. . . . 5 ⊢ (comp‘𝑄) = (comp‘𝑄)
17		evlfcl.a	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ∧ 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌)))
18	17	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
19		evlfcl.b	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐺𝑁𝐻) ∧ 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍)))
20	19	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐺𝑁𝐻))
21	15, 7, 16, 18, 20	fuccocl 17682	. . . 4 ⊢ (𝜑 → (𝐵(⟨𝐹, 𝐺⟩(comp‘𝑄)𝐻)𝐴) ∈ (𝐹𝑁𝐻))
22		eqid 2738	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
23		evlfcl.g	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝑌 ∈ (Base‘𝐶)))
24	23	simprd 496	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
25	17	simprd 496	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝑋(Hom ‘𝐶)𝑌))
26	19	simprd 496	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (𝑌(Hom ‘𝐶)𝑍))
27	4, 5, 22, 2, 12, 24, 13, 25, 26	catcocl 17394	. . . 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 17937	. . 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 17681	. . . 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 7290	. . 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 17577	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
32		1st2ndbr 7883	. . . . . . 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 17586	. . . . 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 7291	. . . 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 17667	. . . . . . . . 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 17671	. . . . . . . 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 7291	. . . . . . 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 2738	. . . . . . . 8 ⊢ (Base‘𝐷) = (Base‘𝐷)
40		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
41	4, 39, 33	funcf1 17581	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
42	41, 24	ffvelrnd 6962	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ (Base‘𝐷))
43	41, 13	ffvelrnd 6962	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ (Base‘𝐷))
44	23	simpld 495	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
45		1st2ndbr 7883	. . . . . . . . . . 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 17581	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
48	47, 13	ffvelrnd 6962	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑍) ∈ (Base‘𝐷))
49	4, 5, 40, 33, 24, 13	funcf2 17583	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(2^nd ‘𝐹)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑍)))
50	49, 26	ffvelrnd 6962	. . . . . . . 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 17669	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑍) ∈ (((1^st ‘𝐹)‘𝑍)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
52		1st2ndbr 7883	. . . . . . . . . . 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 17581	. . . . . . . . 9 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
55	54, 13	ffvelrnd 6962	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐻)‘𝑍) ∈ (Base‘𝐷))
56	7, 20	nat1st2nd 17667	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
57	7, 56, 4, 40, 13	natcl 17669	. . . . . . . 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 17395	. . . . . . 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	ffvelrnd 6962	. . . . . . . 8 ⊢ (𝜑 → ((1^st ‘𝐺)‘𝑌) ∈ (Base‘𝐷))
60	7, 36, 4, 40, 24	natcl 17669	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑌)))
61	4, 5, 40, 46, 24, 13	funcf2 17583	. . . . . . . . 9 ⊢ (𝜑 → (𝑌(2^nd ‘𝐺)𝑍):(𝑌(Hom ‘𝐶)𝑍)⟶(((1^st ‘𝐺)‘𝑌)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑍)))
62	61, 26	ffvelrnd 6962	. . . . . . . 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 17395	. . . . . . 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 2788	. . . . . 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 7290	. . . . 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	ffvelrnd 6962	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑋) ∈ (Base‘𝐷))
67	4, 5, 40, 33, 12, 24	funcf2 17583	. . . . . . 7 ⊢ (𝜑 → (𝑋(2^nd ‘𝐹)𝑌):(𝑋(Hom ‘𝐶)𝑌)⟶(((1^st ‘𝐹)‘𝑋)(Hom ‘𝐷)((1^st ‘𝐹)‘𝑌)))
68	67, 25	ffvelrnd 6962	. . . . . 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 17394	. . . . . 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 17395	. . . . 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 17394	. . . . . 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 17395	. . . . 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 2786	. . . 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 2778	. . 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 2782	. 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 2738	. . . . 5 ⊢ (𝑄 ×_c 𝐶) = (𝑄 ×_c 𝐶)
77	15	fucbas 17677	. . . . 5 ⊢ (𝐶 Func 𝐷) = (Base‘𝑄)
78	15, 7	fuchom 17678	. . . . 5 ⊢ 𝑁 = (Hom ‘𝑄)
79		eqid 2738	. . . . 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 17904	. . . 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 6778	. . 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 7278	. . 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 2796	. 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 7278	. . . . . 6 ⊢ (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩)
85	1, 2, 3, 4, 9, 12	evlf1 17938	. . . . . 6 ⊢ (𝜑 → (𝐹(1^st ‘𝐸)𝑋) = ((1^st ‘𝐹)‘𝑋))
86	84, 85	eqtr3id 2792	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩) = ((1^st ‘𝐹)‘𝑋))
87		df-ov 7278	. . . . . 6 ⊢ (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)
88	1, 2, 3, 4, 44, 24	evlf1 17938	. . . . . 6 ⊢ (𝜑 → (𝐺(1^st ‘𝐸)𝑌) = ((1^st ‘𝐺)‘𝑌))
89	87, 88	eqtr3id 2792	. . . . 5 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩) = ((1^st ‘𝐺)‘𝑌))
90	86, 89	opeq12d 4812	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐸)‘⟨𝐹, 𝑋⟩), ((1^st ‘𝐸)‘⟨𝐺, 𝑌⟩)⟩ = ⟨((1^st ‘𝐹)‘𝑋), ((1^st ‘𝐺)‘𝑌)⟩)
91		df-ov 7278	. . . . 5 ⊢ (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩)
92	1, 2, 3, 4, 11, 13	evlf1 17938	. . . . 5 ⊢ (𝜑 → (𝐻(1^st ‘𝐸)𝑍) = ((1^st ‘𝐻)‘𝑍))
93	91, 92	eqtr3id 2792	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐸)‘⟨𝐻, 𝑍⟩) = ((1^st ‘𝐻)‘𝑍))
94	90, 93	oveq12d 7293	. . 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 7278	. . . 4 ⊢ (𝐵(⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)𝐿) = ((⟨𝐺, 𝑌⟩(2^nd ‘𝐸)⟨𝐻, 𝑍⟩)‘⟨𝐵, 𝐿⟩)
96		eqid 2738	. . . . 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 17937	. . . 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 2792	. . 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 7278	. . . 4 ⊢ (𝐴(⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)𝐾) = ((⟨𝐹, 𝑋⟩(2^nd ‘𝐸)⟨𝐺, 𝑌⟩)‘⟨𝐴, 𝐾⟩)
100		eqid 2738	. . . . 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 17937	. . . 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 2792	. . 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 7296	. 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 2788	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 396 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 class class class wbr 5074 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 Hom chom 16973 compcco 16974 Catccat 17373 Func cfunc 17569 Nat cnat 17657 FuncCat cfuc 17658 ×_c cxpc 17885 eval_F cevlf 17927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-cat 17377 df-func 17573 df-nat 17659 df-fuc 17660 df-xpc 17889 df-evlf 17931

This theorem is referenced by: evlfcl 17940

W3C validator