Theorem arwass 17705

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 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2738	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2738	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		arwlid.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5		arwlid.h	. . . . . . 7 ⊢ 𝐻 = (Homa‘𝐶)
6	5	homarcl 17659	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
7	4, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	5, 1	homarcl2 17666	. . . . . . 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 17666	. . . . . . 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 17670	. . . . . 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 17670	. . . . . 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 17670	. . . . . 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 17312	. . . 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 17700	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐾 · 𝐺)) = ((2nd ‘𝐾)(⟨𝑌, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘𝐺)))
27	26	oveq1d 7270	. . . 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 17700	. . . . 5 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)))
29	28	oveq2d 7271	. . . 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 2788	. . 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 4815	. 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 17701	. . 3 ⊢ (𝜑 → (𝐾 · 𝐺) ∈ (𝑌𝐻𝑊))
33	25, 5, 4, 32, 3	coaval 17699	. 2 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = ⟨𝑋, 𝑊, ((2nd ‘(𝐾 · 𝐺))(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑊)(2nd ‘𝐹))⟩)
34	25, 5, 4, 18	coahom 17701	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))
35	25, 5, 34, 12, 3	coaval 17699	. 2 ⊢ (𝜑 → (𝐾 · (𝐺 · 𝐹)) = ⟨𝑋, 𝑊, ((2nd ‘𝐾)(⟨𝑋, 𝑍⟩(comp‘𝐶)𝑊)(2nd ‘(𝐺 · 𝐹)))⟩)
36	31, 33, 35	3eqtr4d 2788	1 ⊢ (𝜑 → ((𝐾 · 𝐺) · 𝐹) = (𝐾 · (𝐺 · 𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564  ⟨cotp 4566  ‘cfv 6418  (class class class)co 7255  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Hom_achoma 17654  Id_acida 17684  comp_accoa 17685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cat 17294  df-doma 17655  df-coda 17656  df-homa 17657  df-arw 17658  df-coa 17687

This theorem is referenced by: (None)