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Theorem hofcllem 18303
Description: Lemma for hofcl 18304. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofcl.m 𝑀 = (HomF𝐶)
hofcl.o 𝑂 = (oppCat‘𝐶)
hofcl.d 𝐷 = (SetCat‘𝑈)
hofcl.c (𝜑𝐶 ∈ Cat)
hofcl.u (𝜑𝑈𝑉)
hofcl.h (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
hofcllem.b 𝐵 = (Base‘𝐶)
hofcllem.h 𝐻 = (Hom ‘𝐶)
hofcllem.x (𝜑𝑋𝐵)
hofcllem.y (𝜑𝑌𝐵)
hofcllem.z (𝜑𝑍𝐵)
hofcllem.w (𝜑𝑊𝐵)
hofcllem.s (𝜑𝑆𝐵)
hofcllem.t (𝜑𝑇𝐵)
hofcllem.m (𝜑𝐾 ∈ (𝑍𝐻𝑋))
hofcllem.n (𝜑𝐿 ∈ (𝑌𝐻𝑊))
hofcllem.p (𝜑𝑃 ∈ (𝑆𝐻𝑍))
hofcllem.q (𝜑𝑄 ∈ (𝑊𝐻𝑇))
Assertion
Ref Expression
hofcllem (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐿)))

Proof of Theorem hofcllem
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofcllem.b . . . . 5 𝐵 = (Base‘𝐶)
2 hofcllem.h . . . . 5 𝐻 = (Hom ‘𝐶)
3 eqid 2737 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 hofcl.c . . . . . 6 (𝜑𝐶 ∈ Cat)
54adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ Cat)
6 hofcllem.s . . . . . 6 (𝜑𝑆𝐵)
76adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑆𝐵)
8 hofcllem.z . . . . . 6 (𝜑𝑍𝐵)
98adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑍𝐵)
10 hofcllem.x . . . . . 6 (𝜑𝑋𝐵)
1110adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋𝐵)
12 hofcllem.p . . . . . 6 (𝜑𝑃 ∈ (𝑆𝐻𝑍))
1312adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑃 ∈ (𝑆𝐻𝑍))
14 hofcllem.m . . . . . 6 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1514adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐾 ∈ (𝑍𝐻𝑋))
16 hofcllem.t . . . . . 6 (𝜑𝑇𝐵)
1716adantr 480 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑇𝐵)
18 hofcllem.y . . . . . . 7 (𝜑𝑌𝐵)
1918adantr 480 . . . . . 6 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌𝐵)
20 simpr 484 . . . . . 6 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
21 hofcllem.w . . . . . . . 8 (𝜑𝑊𝐵)
22 hofcllem.n . . . . . . . 8 (𝜑𝐿 ∈ (𝑌𝐻𝑊))
23 hofcllem.q . . . . . . . 8 (𝜑𝑄 ∈ (𝑊𝐻𝑇))
241, 2, 3, 4, 18, 21, 16, 22, 23catcocl 17728 . . . . . . 7 (𝜑 → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
2524adantr 480 . . . . . 6 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
261, 2, 3, 5, 11, 19, 17, 20, 25catcocl 17728 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → ((𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑇)𝑓) ∈ (𝑋𝐻𝑇))
271, 2, 3, 5, 7, 9, 11, 13, 15, 17, 26catass 17729 . . . 4 ((𝜑𝑓 ∈ (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)))
2821adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑊𝐵)
2922adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐿 ∈ (𝑌𝐻𝑊))
3023adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑄 ∈ (𝑊𝐻𝑇))
311, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30catass 17729 . . . . . . 7 ((𝜑𝑓 ∈ (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)))
3231oveq1d 7446 . . . . . 6 ((𝜑𝑓 ∈ (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))
331, 2, 3, 5, 11, 19, 28, 20, 29catcocl 17728 . . . . . . 7 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓) ∈ (𝑋𝐻𝑊))
341, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30catass 17729 . . . . . 6 ((𝜑𝑓 ∈ (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)))
3532, 34eqtrd 2777 . . . . 5 ((𝜑𝑓 ∈ (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)))
3635oveq1d 7446 . . . 4 ((𝜑𝑓 ∈ (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))
3727, 36eqtr3d 2779 . . 3 ((𝜑𝑓 ∈ (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))
3837mpteq2dva 5242 . 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 = (HomF𝐶)
401, 2, 3, 4, 6, 8, 10, 12, 14catcocl 17728 . . 3 (𝜑 → (𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃) ∈ (𝑆𝐻𝑋))
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 18301 . 2 (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd𝑀)⟨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))))
4239, 4, 1, 2, 8, 21, 6, 16, 3, 12, 23hof2val 18301 . . . 4 (𝜑 → (𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄) = (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)))
4339, 4, 1, 2, 10, 18, 8, 21, 3, 14, 22hof2val 18301 . . . 4 (𝜑 → (𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐿) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)))
4442, 43oveq12d 7449 . . 3 (𝜑 → ((𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨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 (𝜑𝑈𝑉)
47 eqid 2737 . . . 4 (comp‘𝐷) = (comp‘𝐷)
48 eqid 2737 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4948, 1, 2, 10, 18homfval 17735 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
5048, 1homffn 17736 . . . . . . . 8 (Homf𝐶) Fn (𝐵 × 𝐵)
5150a1i 11 . . . . . . 7 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
52 hofcl.h . . . . . . 7 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
53 df-f 6565 . . . . . . 7 ((Homf𝐶):(𝐵 × 𝐵)⟶𝑈 ↔ ((Homf𝐶) Fn (𝐵 × 𝐵) ∧ ran (Homf𝐶) ⊆ 𝑈))
5451, 52, 53sylanbrc 583 . . . . . 6 (𝜑 → (Homf𝐶):(𝐵 × 𝐵)⟶𝑈)
5554, 10, 18fovcdmd 7605 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) ∈ 𝑈)
5649, 55eqeltrrd 2842 . . . 4 (𝜑 → (𝑋𝐻𝑌) ∈ 𝑈)
5748, 1, 2, 8, 21homfval 17735 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) = (𝑍𝐻𝑊))
5854, 8, 21fovcdmd 7605 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) ∈ 𝑈)
5957, 58eqeltrrd 2842 . . . 4 (𝜑 → (𝑍𝐻𝑊) ∈ 𝑈)
6048, 1, 2, 6, 16homfval 17735 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) = (𝑆𝐻𝑇))
6154, 6, 16fovcdmd 7605 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) ∈ 𝑈)
6260, 61eqeltrrd 2842 . . . 4 (𝜑 → (𝑆𝐻𝑇) ∈ 𝑈)
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 17728 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) ∈ (𝑍𝐻𝑊))
6463fmpttd 7135 . . . 4 (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)):(𝑋𝐻𝑌)⟶(𝑍𝐻𝑊))
654adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝐶 ∈ Cat)
666adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑆𝐵)
678adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑍𝐵)
6816adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑇𝐵)
6912adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑃 ∈ (𝑆𝐻𝑍))
7021adantr 480 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑊𝐵)
71 simpr 484 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑔 ∈ (𝑍𝐻𝑊))
7223adantr 480 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑄 ∈ (𝑊𝐻𝑇))
731, 2, 3, 65, 67, 70, 68, 71, 72catcocl 17728 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) ∈ (𝑍𝐻𝑇))
741, 2, 3, 65, 66, 67, 68, 69, 73catcocl 17728 . . . . 5 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) ∈ (𝑆𝐻𝑇))
7574fmpttd 7135 . . . 4 (𝜑 → (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)):(𝑍𝐻𝑊)⟶(𝑆𝐻𝑇))
7645, 46, 47, 56, 59, 62, 64, 75setcco 18128 . . 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 2738 . . . 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 2738 . . . 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 7439 . . . . 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)))
8079oveq1d 7446 . . . 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))
8163, 77, 78, 80fmptco 7149 . . 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)))
8244, 76, 813eqtrd 2781 . 2 (𝜑 → ((𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨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)))
8338, 41, 823eqtr4d 2787 1 (𝜑 → ((𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃)(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑆, 𝑇⟩)(𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿)) = ((𝑃(⟨𝑍, 𝑊⟩(2nd𝑀)⟨𝑆, 𝑇⟩)𝑄)(⟨(𝑋𝐻𝑌), (𝑍𝐻𝑊)⟩(comp‘𝐷)(𝑆𝐻𝑇))(𝐾(⟨𝑋, 𝑌⟩(2nd𝑀)⟨𝑍, 𝑊⟩)𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  cop 4632  cmpt 5225   × cxp 5683  ran crn 5686  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Homf chomf 17709  oppCatcoppc 17754  SetCatcsetc 18120  HomFchof 18293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-cat 17711  df-homf 17713  df-setc 18121  df-hof 18295
This theorem is referenced by:  hofcl  18304
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