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Theorem hofcllem 17119
Description: Lemma for hofcl 17120. (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 2760 . . . . 5 (comp‘𝐶) = (comp‘𝐶)
4 hofcl.c . . . . . 6 (𝜑𝐶 ∈ Cat)
54adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ Cat)
6 hofcllem.s . . . . . 6 (𝜑𝑆𝐵)
76adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑆𝐵)
8 hofcllem.z . . . . . 6 (𝜑𝑍𝐵)
98adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑍𝐵)
10 hofcllem.x . . . . . 6 (𝜑𝑋𝐵)
1110adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋𝐵)
12 hofcllem.p . . . . . 6 (𝜑𝑃 ∈ (𝑆𝐻𝑍))
1312adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑃 ∈ (𝑆𝐻𝑍))
14 hofcllem.m . . . . . 6 (𝜑𝐾 ∈ (𝑍𝐻𝑋))
1514adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐾 ∈ (𝑍𝐻𝑋))
16 hofcllem.t . . . . . 6 (𝜑𝑇𝐵)
1716adantr 472 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑇𝐵)
18 hofcllem.y . . . . . . 7 (𝜑𝑌𝐵)
1918adantr 472 . . . . . 6 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌𝐵)
20 simpr 479 . . . . . 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 16567 . . . . . . 7 (𝜑 → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
2524adantr 472 . . . . . 6 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝑄(⟨𝑌, 𝑊⟩(comp‘𝐶)𝑇)𝐿) ∈ (𝑌𝐻𝑇))
261, 2, 3, 5, 11, 19, 17, 20, 25catcocl 16567 . . . . 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 16568 . . . 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 472 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑊𝐵)
2922adantr 472 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝐿 ∈ (𝑌𝐻𝑊))
3023adantr 472 . . . . . . . 8 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → 𝑄 ∈ (𝑊𝐻𝑇))
311, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30catass 16568 . . . . . . 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 6829 . . . . . 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 16567 . . . . . . 7 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓) ∈ (𝑋𝐻𝑊))
341, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30catass 16568 . . . . . 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 2794 . . . . 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 6829 . . . 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 2796 . . 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 4896 . 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 16567 . . 3 (𝜑 → (𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃) ∈ (𝑆𝐻𝑋))
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 17117 . 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 17117 . . . 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 17117 . . . 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 6832 . . 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 2760 . . . 4 (comp‘𝐷) = (comp‘𝐷)
48 eqid 2760 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4948, 1, 2, 10, 18homfval 16573 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
5048, 1homffn 16574 . . . . . . . 8 (Homf𝐶) Fn (𝐵 × 𝐵)
5150a1i 11 . . . . . . 7 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
52 hofcl.h . . . . . . 7 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
53 df-f 6053 . . . . . . 7 ((Homf𝐶):(𝐵 × 𝐵)⟶𝑈 ↔ ((Homf𝐶) Fn (𝐵 × 𝐵) ∧ ran (Homf𝐶) ⊆ 𝑈))
5451, 52, 53sylanbrc 701 . . . . . 6 (𝜑 → (Homf𝐶):(𝐵 × 𝐵)⟶𝑈)
5554, 10, 18fovrnd 6972 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) ∈ 𝑈)
5649, 55eqeltrrd 2840 . . . 4 (𝜑 → (𝑋𝐻𝑌) ∈ 𝑈)
5748, 1, 2, 8, 21homfval 16573 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) = (𝑍𝐻𝑊))
5854, 8, 21fovrnd 6972 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) ∈ 𝑈)
5957, 58eqeltrrd 2840 . . . 4 (𝜑 → (𝑍𝐻𝑊) ∈ 𝑈)
6048, 1, 2, 6, 16homfval 16573 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) = (𝑆𝐻𝑇))
6154, 6, 16fovrnd 6972 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) ∈ 𝑈)
6260, 61eqeltrrd 2840 . . . 4 (𝜑 → (𝑆𝐻𝑇) ∈ 𝑈)
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 16567 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) ∈ (𝑍𝐻𝑊))
64 eqid 2760 . . . . 5 (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))
6563, 64fmptd 6549 . . . 4 (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)):(𝑋𝐻𝑌)⟶(𝑍𝐻𝑊))
664adantr 472 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝐶 ∈ Cat)
676adantr 472 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑆𝐵)
688adantr 472 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑍𝐵)
6916adantr 472 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑇𝐵)
7012adantr 472 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑃 ∈ (𝑆𝐻𝑍))
7121adantr 472 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑊𝐵)
72 simpr 479 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑔 ∈ (𝑍𝐻𝑊))
7323adantr 472 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑄 ∈ (𝑊𝐻𝑇))
741, 2, 3, 66, 68, 71, 69, 72, 73catcocl 16567 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) ∈ (𝑍𝐻𝑇))
751, 2, 3, 66, 67, 68, 69, 70, 74catcocl 16567 . . . . 5 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) ∈ (𝑆𝐻𝑇))
76 eqid 2760 . . . . 5 (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))
7775, 76fmptd 6549 . . . 4 (𝜑 → (𝑔 ∈ (𝑍𝐻𝑊) ↦ ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃)):(𝑍𝐻𝑊)⟶(𝑆𝐻𝑇))
7845, 46, 47, 56, 59, 62, 65, 77setcco 16954 . . 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))))
79 eqidd 2761 . . . 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)))
80 eqidd 2761 . . . 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)))
81 oveq2 6822 . . . . 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)))
8281oveq1d 6829 . . . 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))
8363, 79, 80, 82fmptco 6560 . . 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)))
8444, 78, 833eqtrd 2798 . 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)))
8538, 41, 843eqtr4d 2804 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 383   = wceq 1632  wcel 2139  wss 3715  cop 4327  cmpt 4881   × cxp 5264  ran crn 5267  ccom 5270   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  2nd c2nd 7333  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546  Homf chomf 16548  oppCatcoppc 16592  SetCatcsetc 16946  HomFchof 17109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-hom 16188  df-cco 16189  df-cat 16550  df-homf 16552  df-setc 16947  df-hof 17111
This theorem is referenced by:  hofcl  17120
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