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Theorem hofcllem 18314
Description: Lemma for hofcl 18315. (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 2734 . . . . 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 17729 . . . . . . 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 17729 . . . . 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 17730 . . . 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 17730 . . . . . . 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 7445 . . . . . 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 17729 . . . . . . 7 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓) ∈ (𝑋𝐻𝑊))
341, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30catass 17730 . . . . . 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 2774 . . . . 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 7445 . . . 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 2776 . . 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 5247 . 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 17729 . . 3 (𝜑 → (𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃) ∈ (𝑆𝐻𝑋))
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 18312 . 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 18312 . . . 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 18312 . . . 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 7448 . . 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 2734 . . . 4 (comp‘𝐷) = (comp‘𝐷)
48 eqid 2734 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4948, 1, 2, 10, 18homfval 17736 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
5048, 1homffn 17737 . . . . . . . 8 (Homf𝐶) Fn (𝐵 × 𝐵)
5150a1i 11 . . . . . . 7 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
52 hofcl.h . . . . . . 7 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
53 df-f 6566 . . . . . . 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 7604 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) ∈ 𝑈)
5649, 55eqeltrrd 2839 . . . 4 (𝜑 → (𝑋𝐻𝑌) ∈ 𝑈)
5748, 1, 2, 8, 21homfval 17736 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) = (𝑍𝐻𝑊))
5854, 8, 21fovcdmd 7604 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) ∈ 𝑈)
5957, 58eqeltrrd 2839 . . . 4 (𝜑 → (𝑍𝐻𝑊) ∈ 𝑈)
6048, 1, 2, 6, 16homfval 17736 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) = (𝑆𝐻𝑇))
6154, 6, 16fovcdmd 7604 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) ∈ 𝑈)
6260, 61eqeltrrd 2839 . . . 4 (𝜑 → (𝑆𝐻𝑇) ∈ 𝑈)
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 17729 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) ∈ (𝑍𝐻𝑊))
6463fmpttd 7134 . . . 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 17729 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) ∈ (𝑍𝐻𝑇))
741, 2, 3, 65, 66, 67, 68, 69, 73catcocl 17729 . . . . 5 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) ∈ (𝑆𝐻𝑇))
7574fmpttd 7134 . . . 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 18136 . . 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 2735 . . . 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 2735 . . . 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 7438 . . . . 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 7445 . . . 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 7148 . . 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 2778 . 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 2784 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 1536  wcel 2105  wss 3962  cop 4636  cmpt 5230   × cxp 5686  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  2nd c2nd 8011  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Homf chomf 17710  oppCatcoppc 17755  SetCatcsetc 18128  HomFchof 18304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-cat 17712  df-homf 17714  df-setc 18129  df-hof 18306
This theorem is referenced by:  hofcl  18315
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