Theorem coahom 16543

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 2610	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	coaval 16541	. 2 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩)
7		eqid 2610	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8	2	homarcl 16501	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		eqid 2610	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11	2, 7	homarcl2 16508	. . . . 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 474	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
14	2, 7	homarcl2 16508	. . . . 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 478	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
17	12	simprd 478	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
18	2, 10	homahom 16512	. . . . 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 16512	. . . . 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 16169	. . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
23	2, 7, 9, 10, 13, 16, 22	elhomai2 16507	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩ ∈ (𝑋𝐻𝑍))
24	6, 23	eqeltrd 2688	1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ⟨cotp 4133  ‘cfv 5804  (class class class)co 6549  2^nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Hom_achoma 16496  comp_accoa 16527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-doma 16497  df-coda 16498  df-homa 16499  df-arw 16500  df-coa 16529

This theorem is referenced by:  coapm  16544  arwass  16547