Theorem coaval 18249

Description: Value of composition for composable arrows. (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))
coaval.x	⊢ ∙ = (comp‘𝐶)

Assertion

Ref	Expression
coaval	⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)

Proof of Theorem coaval

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homdmcoa.o	. . 3 ⊢ · = (compa‘𝐶)
2		eqid 2760	. . 3 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
3		coaval.x	. . 3 ⊢ ∙ = (comp‘𝐶)
4	1, 2, 3	coafval 18245	. 2 ⊢ · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩)
5		homdmcoa.h	. . . . 5 ⊢ 𝐻 = (Homa‘𝐶)
6	2, 5	homarw 18227	. . . 4 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3906	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9		fveqeq2 6895	. . . 4 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘𝐹) = (doma‘𝑔)))
10	2, 5	homarw 18227	. . . . 5 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
11		homdmcoa.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
13	10, 12	sselid 3906	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
14	5	homacd 18222	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
15	12, 14	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐹) = 𝑌)
16		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
17	16	fveq2d 6890	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = (doma‘𝐺))
18	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
19	5	homadm 18221	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
20	18, 19	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐺) = 𝑌)
21	17, 20	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = 𝑌)
22	15, 21	eqtr4d 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐹) = (doma‘𝑔))
23	9, 13, 22	elrabd 3646	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (coda‘ℎ) = (doma‘𝑔)})
24		otex 5433	. . . 4 ⊢ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V
25	24	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩ ∈ V)
26		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
27	26	fveq2d 6890	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = (doma‘𝐹))
28	5	homadm 18221	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋)
29	12, 28	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐹) = 𝑋)
30	29	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝐹) = 𝑋)
31	27, 30	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = 𝑋)
32	16	fveq2d 6890	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = (coda‘𝐺))
33	5	homacd 18222	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (coda‘𝐺) = 𝑍)
34	18, 33	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐺) = 𝑍)
35	32, 34	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = 𝑍)
36	35	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (coda‘𝑔) = 𝑍)
37	21	adantrr 730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑔) = 𝑌)
38	31, 37	opeq12d 4847	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(doma‘𝑓), (doma‘𝑔)⟩ = ⟨𝑋, 𝑌⟩)
39	38, 36	oveq12d 7439	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔)) = (⟨𝑋, 𝑌⟩ ∙ 𝑍))
40		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
41	40	fveq2d 6890	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑔) = (2nd ‘𝐺))
42	26	fveq2d 6890	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑓) = (2nd ‘𝐹))
43	39, 41, 42	oveq123d 7442	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))
44	31, 36, 43	oteq123d 4865	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)
45	8, 23, 25, 44	ovmpodv2 7579	. 2 ⊢ (𝜑 → ( · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔))(2nd ‘𝑓))⟩) → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩))
46	4, 45	mpi 21	1 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450  ⟨cop 4834  ⟨cotp 4857  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  2^nd c2nd 8001  compcco 17446  dom_acdoma 18201  cod_accoda 18202  Arrowcarw 18203  Hom_achoma 18204  comp_accoa 18235

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 7752

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 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-doma 18205  df-coda 18206  df-homa 18207  df-arw 18208  df-coa 18237

This theorem is used by:  coa2  18250  coahom  18251  arwlid  18253  arwrid  18254  arwass  18255