Theorem arwass 18063

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 2728	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2728	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2728	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		arwlid.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5		arwlid.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
6	5	homarcl 18017	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	5, 1	homarcl2 18024	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
9	4, 8	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
10	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	9	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		arwass.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))
13	5, 1	homarcl2 18024	. . . . . . 7 ⊢ (𝐾 ∈ (𝑍𝐻𝑊) → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑍 ∈ (Base‘𝐶) ∧ 𝑊 ∈ (Base‘𝐶)))
15	14	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
16	5, 2	homahom 18028	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
17	4, 16	syl 17	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
18		arwass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
19	5, 2	homahom 18028	. . . . . 6 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
20	18, 19	syl 17	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
21	14	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐶))
22	5, 2	homahom 18028	. . . . . 6 ⊢ (𝐾 ∈ (𝑍𝐻𝑊) → (2nd ‘𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
23	12, 22	syl 17	. . . . 5 ⊢ (𝜑 → (2nd ‘𝐾) ∈ (𝑍(Hom ‘𝐶)𝑊))
24	1, 2, 3, 7, 10, 11, 15, 17, 20, 21, 23	catass 17666	. . . 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 18058	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺)))
27	26	oveq1d 7435	. . . 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 18058	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)))
29	28	oveq2d 7436	. . . 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 2778	. . 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 4888	. 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 18059	. . 3 ⊢ (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
33	25, 5, 4, 32, 3	coaval 18057	. 2 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹))⟩)
34	25, 5, 4, 18	coahom 18059	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
35	25, 5, 34, 12, 3	coaval 18057	. 2 ⊢ (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
36	31, 33, 35	3eqtr4d 2778	1 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ⟨cop 4635  ⟨cotp 4637  ‘cfv 6548  (class class class)co 7420  2^nd c2nd 7992  Basecbs 17180  Hom chom 17244  compcco 17245  Catccat 17644  Hom_achoma 18012  Id_acida 18042  comp_accoa 18043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-cat 17648  df-doma 18013  df-coda 18014  df-homa 18015  df-arw 18016  df-coa 18045

This theorem is referenced by: (None)