Theorem coahom 18123

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 2734	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	coaval 18121	. 2 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩)
7		eqid 2734	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8	2	homarcl 18081	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		eqid 2734	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11	2, 7	homarcl2 18088	. . . . 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 494	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
14	2, 7	homarcl2 18088	. . . . 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 495	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
17	12	simprd 495	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
18	2, 10	homahom 18092	. . . . 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 18092	. . . . 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 17729	. . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
23	2, 7, 9, 10, 13, 16, 22	elhomai2 18087	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩ ∈ (𝑋𝐻𝑍))
24	6, 23	eqeltrd 2838	1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ⟨cop 4636  ⟨cotp 4638  ‘cfv 6562  (class class class)co 7430  2^nd c2nd 8011  Basecbs 17244  Hom chom 17308  compcco 17309  Catccat 17708  Hom_achoma 18076  comp_accoa 18107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-cat 17712  df-doma 18077  df-coda 18078  df-homa 18079  df-arw 18080  df-coa 18109

This theorem is referenced by:  coapm  18124  arwass  18127