Theorem coaval 18051

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 2728	. . 3 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
3		coaval.x	. . 3 ⊢ ∙ = (comp‘𝐶)
4	1, 2, 3	coafval 18047	. 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 18029	. . . 4 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3977	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9		fveqeq2 6901	. . . 4 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘𝐹) = (doma‘𝑔)))
10	2, 5	homarw 18029	. . . . 5 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
11		homdmcoa.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
13	10, 12	sselid 3977	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
14	5	homacd 18024	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
15	12, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐹) = 𝑌)
16		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
17	16	fveq2d 6896	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = (doma‘𝐺))
18	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
19	5	homadm 18023	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐺) = 𝑌)
21	17, 20	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = 𝑌)
22	15, 21	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐹) = (doma‘𝑔))
23	9, 13, 22	elrabd 3683	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (coda‘ℎ) = (doma‘𝑔)})
24		otex 5462	. . . 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 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
27	26	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = (doma‘𝐹))
28	5	homadm 18023	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋)
29	12, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐹) = 𝑋)
30	29	adantrr 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝐹) = 𝑋)
31	27, 30	eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = 𝑋)
32	16	fveq2d 6896	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = (coda‘𝐺))
33	5	homacd 18024	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (coda‘𝐺) = 𝑍)
34	18, 33	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐺) = 𝑍)
35	32, 34	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = 𝑍)
36	35	adantrr 716	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (coda‘𝑔) = 𝑍)
37	21	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑔) = 𝑌)
38	31, 37	opeq12d 4878	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(doma‘𝑓), (doma‘𝑔)⟩ = ⟨𝑋, 𝑌⟩)
39	38, 36	oveq12d 7433	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(doma‘𝑓), (doma‘𝑔)⟩ ∙ (coda‘𝑔)) = (⟨𝑋, 𝑌⟩ ∙ 𝑍))
40		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
41	40	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑔) = (2nd ‘𝐺))
42	26	fveq2d 6896	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑓) = (2nd ‘𝐹))
43	39, 41, 42	oveq123d 7436	. . . 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 4885	. . 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 7574	. 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 20	1 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {crab 3428  Vcvv 3470  ⟨cop 4631  ⟨cotp 4633  ‘cfv 6543  (class class class)co 7415   ∈ cmpo 7417  2^nd c2nd 7987  compcco 17239  dom_acdoma 18003  cod_accoda 18004  Arrowcarw 18005  Hom_achoma 18006  comp_accoa 18037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-ot 4634  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-doma 18007  df-coda 18008  df-homa 18009  df-arw 18010  df-coa 18039

This theorem is referenced by:  coa2  18052  coahom  18053  arwlid  18055  arwrid  18056  arwass  18057