Theorem coahom 18026

Description: The composition of two composable arrows is an arrow. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homdmcoa.o	⊢ · = (compa‘𝐶)
homdmcoa.h	⊢ 𝐻 = (Homa‘𝐶)
homdmcoa.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))

Assertion

Ref	Expression
coahom	⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Proof of Theorem coahom

Step	Hyp	Ref	Expression
1		homdmcoa.o	. . 3 ⊢ · = (compa‘𝐶)
2		homdmcoa.h	. . 3 ⊢ 𝐻 = (Homa‘𝐶)
3		homdmcoa.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		homdmcoa.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
5		eqid 2730	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	coaval 18024	. 2 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩)
7		eqid 2730	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8	2	homarcl 17984	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		eqid 2730	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11	2, 7	homarcl2 17991	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
12	3, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
13	12	simpld 493	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
14	2, 7	homarcl2 17991	. . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
15	4, 14	syl 17	. . . 4 ⊢ (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
16	15	simprd 494	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
17	12	simprd 494	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
18	2, 10	homahom 17995	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
19	3, 18	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
20	2, 10	homahom 17995	. . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
21	4, 20	syl 17	. . . 4 ⊢ (𝜑 → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
22	7, 10, 5, 9, 13, 17, 16, 19, 21	catcocl 17635	. . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
23	2, 7, 9, 10, 13, 16, 22	elhomai2 17990	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩ ∈ (𝑋𝐻𝑍))
24	6, 23	eqeltrd 2831	1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ⟨cop 4635  ⟨cotp 4637  ‘cfv 6544  (class class class)co 7413  2^nd c2nd 7978  Basecbs 17150  Hom chom 17214  compcco 17215  Catccat 17614  Hom_achoma 17979  comp_accoa 18010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7979  df-2nd 7980  df-cat 17618  df-doma 17980  df-coda 17981  df-homa 17982  df-arw 17983  df-coa 18012

This theorem is referenced by:  coapm  18027  arwass  18030