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 2737	. . 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 2737	. . 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 2737	. . 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 494	. . 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 495	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
17	12	simprd 495	. . . 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 17640	. . 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 2837	1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  ⟨cotp 4576  ‘cfv 6490  (class class class)co 7358  2^nd c2nd 7932  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-ot 4577  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-cat 17623  df-doma 17980  df-coda 17981  df-homa 17982  df-arw 17983  df-coa 18012

This theorem is referenced by:  coapm  18027  arwass  18030