Theorem arwass 18036

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 2741	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2741	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2741	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		arwlid.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5		arwlid.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
6	5	homarcl 17990	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	5, 1	homarcl2 17997	. . . . . . 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 496	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
11	9	simprd 497	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
12		arwass.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑊))
13	5, 1	homarcl2 17997	. . . . . . 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 496	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
16	5, 2	homahom 18001	. . . . . 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 18001	. . . . . 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 497	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐶))
22	5, 2	homahom 18001	. . . . . 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 17647	. . . 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 18031	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺)))
27	26	oveq1d 7375	. . . 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 18031	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)))
29	28	oveq2d 7376	. . . 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 2786	. . 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 4821	. 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 18032	. . 3 ⊢ (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
33	25, 5, 4, 32, 3	coaval 18030	. 2 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹))⟩)
34	25, 5, 4, 18	coahom 18032	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
35	25, 5, 34, 12, 3	coaval 18030	. 2 ⊢ (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
36	31, 33, 35	3eqtr4d 2786	1 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ⟨cop 4564  ⟨cotp 4566  ‘cfv 6489  (class class class)co 7360  2^nd c2nd 7934  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625  Hom_achoma 17985  Id_acida 18015  comp_accoa 18016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-cat 17629  df-doma 17986  df-coda 17987  df-homa 17988  df-arw 17989  df-coa 18018

This theorem is referenced by: (None)