Theorem coahom 18250

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 2760	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	coaval 18248	. 2 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩)
7		eqid 2760	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
8	2	homarcl 18208	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
9	3, 8	syl 18	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		eqid 2760	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
11	2, 7	homarcl2 18215	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
12	3, 11	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
13	12	simpld 500	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
14	2, 7	homarcl2 18215	. . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
15	4, 14	syl 18	. . . 4 ⊢ (𝜑 → (𝑌 ∈ (Base‘𝐶) ∧ 𝑍 ∈ (Base‘𝐶)))
16	15	simprd 501	. . 3 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐶))
17	12	simprd 501	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶))
18	2, 10	homahom 18219	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
19	3, 18	syl 18	. . . 4 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
20	2, 10	homahom 18219	. . . . 5 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
21	4, 20	syl 18	. . . 4 ⊢ (𝜑 → (2nd ‘𝐺) ∈ (𝑌(Hom ‘𝐶)𝑍))
22	7, 10, 5, 9, 13, 17, 16, 19, 21	catcocl 17864	. . 3 ⊢ (𝜑 → ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹)) ∈ (𝑋(Hom ‘𝐶)𝑍))
23	2, 7, 9, 10, 13, 16, 22	elhomai2 18214	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)(2nd ‘𝐹))⟩ ∈ (𝑋𝐻𝑍))
24	6, 23	eqeltrd 2860	1 ⊢ (𝜑 → (𝐺 · 𝐹) ∈ (𝑋𝐻𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ⟨cop 4834  ⟨cotp 4857  ‘cfv 6538  (class class class)co 7420  2^nd c2nd 8000  Basecbs 17392  Hom chom 17444  compcco 17445  Catccat 17843  Hom_achoma 18203  comp_accoa 18234

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-cat 17847  df-doma 18204  df-coda 18205  df-homa 18206  df-arw 18207  df-coa 18236

This theorem is used by:  coapm  18251  arwass  18254