Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hofcllem Structured version   Visualization version   GIF version

Theorem hofcllem 16721
 Description: Lemma for hofcl 16722. (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 2610 . . . . 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 476 . . . . . 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 16169 . . . . . . 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 16169 . . . . 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 16170 . . . 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 16170 . . . . . . 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 6564 . . . . . 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 16169 . . . . . . 7 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → (𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓) ∈ (𝑋𝐻𝑊))
341, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30catass 16170 . . . . . 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 2644 . . . . 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 6564 . . . 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 2646 . . 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 4672 . 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 16169 . . 3 (𝜑 → (𝐾(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑋)𝑃) ∈ (𝑆𝐻𝑋))
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 16719 . 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 16719 . . . 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 16719 . . . 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 6567 . . 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 2610 . . . 4 (comp‘𝐷) = (comp‘𝐷)
48 eqid 2610 . . . . . 6 (Homf𝐶) = (Homf𝐶)
4948, 1, 2, 10, 18homfval 16175 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
5048, 1homffn 16176 . . . . . . . 8 (Homf𝐶) Fn (𝐵 × 𝐵)
5150a1i 11 . . . . . . 7 (𝜑 → (Homf𝐶) Fn (𝐵 × 𝐵))
52 hofcl.h . . . . . . 7 (𝜑 → ran (Homf𝐶) ⊆ 𝑈)
53 df-f 5808 . . . . . . 7 ((Homf𝐶):(𝐵 × 𝐵)⟶𝑈 ↔ ((Homf𝐶) Fn (𝐵 × 𝐵) ∧ ran (Homf𝐶) ⊆ 𝑈))
5451, 52, 53sylanbrc 695 . . . . . 6 (𝜑 → (Homf𝐶):(𝐵 × 𝐵)⟶𝑈)
5554, 10, 18fovrnd 6704 . . . . 5 (𝜑 → (𝑋(Homf𝐶)𝑌) ∈ 𝑈)
5649, 55eqeltrrd 2689 . . . 4 (𝜑 → (𝑋𝐻𝑌) ∈ 𝑈)
5748, 1, 2, 8, 21homfval 16175 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) = (𝑍𝐻𝑊))
5854, 8, 21fovrnd 6704 . . . . 5 (𝜑 → (𝑍(Homf𝐶)𝑊) ∈ 𝑈)
5957, 58eqeltrrd 2689 . . . 4 (𝜑 → (𝑍𝐻𝑊) ∈ 𝑈)
6048, 1, 2, 6, 16homfval 16175 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) = (𝑆𝐻𝑇))
6154, 6, 16fovrnd 6704 . . . . 5 (𝜑 → (𝑆(Homf𝐶)𝑇) ∈ 𝑈)
6260, 61eqeltrrd 2689 . . . 4 (𝜑 → (𝑆𝐻𝑇) ∈ 𝑈)
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 16169 . . . . 5 ((𝜑𝑓 ∈ (𝑋𝐻𝑌)) → ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾) ∈ (𝑍𝐻𝑊))
64 eqid 2610 . . . . 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 6292 . . . 4 (𝜑 → (𝑓 ∈ (𝑋𝐻𝑌) ↦ ((𝐿(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)𝑓)(⟨𝑍, 𝑋⟩(comp‘𝐶)𝑊)𝐾)):(𝑋𝐻𝑌)⟶(𝑍𝐻𝑊))
664adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝐶 ∈ Cat)
676adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑆𝐵)
688adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑍𝐵)
6916adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑇𝐵)
7012adantr 480 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑃 ∈ (𝑆𝐻𝑍))
7121adantr 480 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑊𝐵)
72 simpr 476 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑔 ∈ (𝑍𝐻𝑊))
7323adantr 480 . . . . . . 7 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → 𝑄 ∈ (𝑊𝐻𝑇))
741, 2, 3, 66, 68, 71, 69, 72, 73catcocl 16169 . . . . . 6 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → (𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔) ∈ (𝑍𝐻𝑇))
751, 2, 3, 66, 67, 68, 69, 70, 74catcocl 16169 . . . . 5 ((𝜑𝑔 ∈ (𝑍𝐻𝑊)) → ((𝑄(⟨𝑍, 𝑊⟩(comp‘𝐶)𝑇)𝑔)(⟨𝑆, 𝑍⟩(comp‘𝐶)𝑇)𝑃) ∈ (𝑆𝐻𝑇))
76 eqid 2610 . . . . 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 6292 . . . 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 16556 . . 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 2611 . . . 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 2611 . . . 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 6557 . . . . 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 6564 . . . 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 6303 . . 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 2648 . 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 2654 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 1475   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  2nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Homf chomf 16150  oppCatcoppc 16194  SetCatcsetc 16548  HomFchof 16711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-homf 16154  df-setc 16549  df-hof 16713 This theorem is referenced by:  hofcl  16722
 Copyright terms: Public domain W3C validator