Theorem hofcllem 17976

Description: Lemma for hofcl 17977. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofcl.m	⊢ 𝑀 = (HomF‘𝐶)
hofcl.o	⊢ 𝑂 = (oppCat‘𝐶)
hofcl.d	⊢ 𝐷 = (SetCat‘𝑈)
hofcl.c	⊢ (𝜑 → 𝐶 ∈ Cat)
hofcl.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
hofcl.h	⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
hofcllem.b	⊢ 𝐵 = (Base‘𝐶)
hofcllem.h	⊢ 𝐻 = (Hom ‘𝐶)
hofcllem.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
hofcllem.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
hofcllem.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
hofcllem.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
hofcllem.s	⊢ (𝜑 → 𝑆 ∈ 𝐵)
hofcllem.t	⊢ (𝜑 → 𝑇 ∈ 𝐵)
hofcllem.m	⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
hofcllem.n	⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑊))
hofcllem.p	⊢ (𝜑 → 𝑃 ∈ (𝑆𝐻𝑍))
hofcllem.q	⊢ (𝜑 → 𝑄 ∈ (𝑊𝐻𝑇))

Assertion

Ref	Expression
hofcllem	⊢ (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿)))

Proof of Theorem hofcllem

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofcllem.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		hofcllem.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2738	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		hofcl.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ Cat)
6		hofcllem.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑆 ∈ 𝐵)
8		hofcllem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑍 ∈ 𝐵)
10		hofcllem.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	10	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
12		hofcllem.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (𝑆𝐻𝑍))
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑃 ∈ (𝑆𝐻𝑍))
14		hofcllem.m	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
15	14	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐾 ∈ (𝑍𝐻𝑋))
16		hofcllem.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
17	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑇 ∈ 𝐵)
18		hofcllem.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
19	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
20		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
21		hofcllem.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
22		hofcllem.n	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (𝑌𝐻𝑊))
23		hofcllem.q	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (𝑊𝐻𝑇))
24	1, 2, 3, 4, 18, 21, 16, 22, 23	catcocl 17394	. . . . . . 7 ⊢ (𝜑 → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
25	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
26	1, 2, 3, 5, 11, 19, 17, 20, 25	catcocl 17394	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓) ∈ (𝑋𝐻𝑇))
27	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 26	catass 17395	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) = (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑆, 𝑋⟩(comp‘𝐶)𝑇)(𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)))
28	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑊 ∈ 𝐵)
29	22	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐿 ∈ (𝑌𝐻𝑊))
30	23	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑄 ∈ (𝑊𝐻𝑇))
31	1, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30	catass 17395	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓) = (𝑄(⟨𝑋, 𝑊⟩(comp‘𝐶)𝑇)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)))
32	31	oveq1d 7290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾) = ((𝑄(⟨𝑋, 𝑊⟩(comp‘𝐶)𝑇)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓))(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾))
33	1, 2, 3, 5, 11, 19, 28, 20, 29	catcocl 17394	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓) ∈ (𝑋𝐻𝑊))
34	1, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30	catass 17395	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((𝑄(⟨𝑋, 𝑊⟩(comp‘𝐶)𝑇)(𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓))(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾) = (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
35	32, 34	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾) = (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
36	35	oveq1d 7290	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑇)𝐾)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) = ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃))
37	27, 36	eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑆, 𝑋⟩(comp‘𝐶)𝑇)(𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)) = ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃))
38	37	mpteq2dva 5174	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑆, 𝑋⟩(comp‘𝐶)𝑇)(𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃))) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
39		hofcl.m	. . 3 ⊢ 𝑀 = (HomF‘𝐶)
40	1, 2, 3, 4, 6, 8, 10, 12, 14	catcocl 17394	. . 3 ⊢ (𝜑 → (𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃) ∈ (𝑆𝐻𝑋))
41	39, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24	hof2val 17974	. 2 ⊢ (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ (((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓)(⟨𝑆, 𝑋⟩(comp‘𝐶)𝑇)(𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃))))
42	39, 4, 1, 2, 8, 21, 6, 16, 3, 12, 23	hof2val 17974	. . . 4 ⊢ (𝜑 → (𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄) = (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
43	39, 4, 1, 2, 10, 18, 8, 21, 3, 14, 22	hof2val 17974	. . . 4 ⊢ (𝜑 → (𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
44	42, 43	oveq12d 7293	. . 3 ⊢ (𝜑 → ((𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿)) = ((𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃))(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))))
45		hofcl.d	. . . 4 ⊢ 𝐷 = (SetCat‘𝑈)
46		hofcl.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
47		eqid 2738	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
48		eqid 2738	. . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶)
49	48, 1, 2, 10, 18	homfval 17401	. . . . 5 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) = (𝑋𝐻𝑌))
50	48, 1	homffn 17402	. . . . . . . 8 ⊢ (Homf ‘𝐶) Fn (𝐵 × 𝐵)
51	50	a1i 11	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐶) Fn (𝐵 × 𝐵))
52		hofcl.h	. . . . . . 7 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈)
53		df-f 6437	. . . . . . 7 ⊢ ((Homf ‘𝐶):(𝐵 × 𝐵)⟶𝑈 ↔ ((Homf ‘𝐶) Fn (𝐵 × 𝐵) ∧ ran (Homf ‘𝐶) ⊆ 𝑈))
54	51, 52, 53	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶):(𝐵 × 𝐵)⟶𝑈)
55	54, 10, 18	fovrnd 7444	. . . . 5 ⊢ (𝜑 → (𝑋(Homf ‘𝐶)𝑌) ∈ 𝑈)
56	49, 55	eqeltrrd 2840	. . . 4 ⊢ (𝜑 → (𝑋𝐻𝑌) ∈ 𝑈)
57	48, 1, 2, 8, 21	homfval 17401	. . . . 5 ⊢ (𝜑 → (𝑍(Homf ‘𝐶)𝑊) = (𝑍𝐻𝑊))
58	54, 8, 21	fovrnd 7444	. . . . 5 ⊢ (𝜑 → (𝑍(Homf ‘𝐶)𝑊) ∈ 𝑈)
59	57, 58	eqeltrrd 2840	. . . 4 ⊢ (𝜑 → (𝑍𝐻𝑊) ∈ 𝑈)
60	48, 1, 2, 6, 16	homfval 17401	. . . . 5 ⊢ (𝜑 → (𝑆(Homf ‘𝐶)𝑇) = (𝑆𝐻𝑇))
61	54, 6, 16	fovrnd 7444	. . . . 5 ⊢ (𝜑 → (𝑆(Homf ‘𝐶)𝑇) ∈ 𝑈)
62	60, 61	eqeltrrd 2840	. . . 4 ⊢ (𝜑 → (𝑆𝐻𝑇) ∈ 𝑈)
63	1, 2, 3, 5, 9, 11, 28, 15, 33	catcocl 17394	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) ∈ (𝑍𝐻𝑊))
64	63	fmpttd 6989	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)):(𝑋𝐻𝑌)⟶(𝑍𝐻𝑊))
65	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝐶 ∈ Cat)
66	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑆 ∈ 𝐵)
67	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑍 ∈ 𝐵)
68	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑇 ∈ 𝐵)
69	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑃 ∈ (𝑆𝐻𝑍))
70	21	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑊 ∈ 𝐵)
71		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑔 ∈ (𝑍𝐻𝑊))
72	23	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → 𝑄 ∈ (𝑊𝐻𝑇))
73	1, 2, 3, 65, 67, 70, 68, 71, 72	catcocl 17394	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) ∈ (𝑍𝐻𝑇))
74	1, 2, 3, 65, 66, 67, 68, 69, 73	catcocl 17394	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑍𝐻𝑊)) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) ∈ (𝑆𝐻𝑇))
75	74	fmpttd 6989	. . . 4 ⊢ (𝜑 → (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)):(𝑍𝐻𝑊)⟶(𝑆𝐻𝑇))
76	45, 46, 47, 56, 59, 62, 64, 75	setcco 17798	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃))(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))) = ((𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)) ∘ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))))
77		eqidd 2739	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
78		eqidd 2739	. . . 4 ⊢ (𝜑 → (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)) = (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
79		oveq2 7283	. . . . 5 ⊢ (𝑔 = ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) = (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
80	79	oveq1d 7290	. . . 4 ⊢ (𝑔 = ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) = ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃))
81	63, 77, 78, 80	fmptco 7001	. . 3 ⊢ (𝜑 → ((𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)) ∘ (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
82	44, 76, 81	3eqtrd 2782	. 2 ⊢ (𝜑 → ((𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿)) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾))(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
83	38, 41, 82	3eqtr4d 2788	1 ⊢ (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd ‘𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐿)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ⟨cop 4567   ↦ cmpt 5157   × cxp 5587  ran crn 5590   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  2^nd c2nd 7830  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Hom_f chomf 17375  oppCatcoppc 17420  SetCatcsetc 17790  Hom_Fchof 17966

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-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-homf 17379  df-setc 17791  df-hof 17968

This theorem is referenced by:  hofcl  17977