Theorem arwass 18275

Description: Associativity of composition in a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwlid.h	⊢ 𝐻 = (Homa‘𝐶)
arwlid.o	⊢ · = (compa‘𝐶)
arwlid.a	⊢ 1 = (Ida‘𝐶)
arwlid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
arwass.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
arwass.k	⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))

Assertion

Ref	Expression
arwass	⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

Proof of Theorem arwass

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2763	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		arwlid.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5		arwlid.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
6	5	homarcl 18229	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 6	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	5, 1	homarcl2 18236	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	4, 8	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simpld 500	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	9	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		arwass.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))
13	5, 1	homarcl2 18236	. . . . . . 7 ⊢ (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
14	12, 13	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
15	14	simpld 500	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
16	5, 2	homahom 18240	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
17	4, 16	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18		arwass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
19	5, 2	homahom 18240	. . . . . 6 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
20	18, 19	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
21	14	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐶))
22	5, 2	homahom 18240	. . . . . 6 ⊢ (𝐾 ∈ (𝑍𝐻𝑊) → (2nd ‘𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
23	12, 22	syl 18	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
24	1, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23	catass 17886	. . . 4 ⊢ (𝜑 → (((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹)) = ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))))
25		arwlid.o	. . . . . 6 ⊢ · = (compa‘𝐶)
26	25, 5, 18, 12, 3	coa2 18270	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺)))
27	26	oveq1d 7455	. . . 4 ⊢ (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹)) = (((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹)))
28	25, 5, 4, 18, 3	coa2 18270	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)))
29	28	oveq2d 7456	. . . 4 ⊢ (𝜑 → ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))) = ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))))
30	24, 27, 29	3eqtr4d 2808	. . 3 ⊢ (𝜑 → ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹)) = ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹))))
31	30	oteq3d 4868	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹))⟩ = ⟨𝑋, 𝑊, ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
32	25, 5, 18, 12	coahom 18271	. . 3 ⊢ (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
33	25, 5, 4, 32, 3	coaval 18269	. 2 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹))⟩)
34	25, 5, 4, 18	coahom 18271	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
35	25, 5, 34, 12, 3	coaval 18269	. 2 ⊢ (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
36	31, 33, 35	3eqtr4d 2808	1 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  ⟨cotp 4861  ‘cfv 6555  (class class class)co 7440  2^nd c2nd 8021  Basecbs 17413  Hom chom 17465  compcco 17466  Catccat 17864  Hom_achoma 18224  Id_acida 18254  comp_accoa 18255

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-cat 17868  df-doma 18225  df-coda 18226  df-homa 18227  df-arw 18228  df-coa 18257

This theorem is used by: (None)