Theorem coaval 17970

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 2731	. . 3 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
3		coaval.x	. . 3 ⊢ ∙ = (comp‘𝐶)
4	1, 2, 3	coafval 17966	. 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 17948	. . . 4 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3927	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9		fveqeq2 6826	. . . 4 ⊢ (ℎ = 𝐹 → ((coda‘ℎ) = (doma‘𝑔) ↔ (coda‘𝐹) = (doma‘𝑔)))
10	2, 5	homarw 17948	. . . . 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 3927	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
14	5	homacd 17943	. . . . . 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 6821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = (doma‘𝐺))
18	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
19	5	homadm 17942	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐺) = 𝑌)
21	17, 20	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝑔) = 𝑌)
22	15, 21	eqtr4d 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐹) = (doma‘𝑔))
23	9, 13, 22	elrabd 3644	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (coda‘ℎ) = (doma‘𝑔)})
24		otex 5400	. . . 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 6821	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = (doma‘𝐹))
28	5	homadm 17942	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (doma‘𝐹) = 𝑋)
29	12, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (doma‘𝐹) = 𝑋)
30	29	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝐹) = 𝑋)
31	27, 30	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑓) = 𝑋)
32	16	fveq2d 6821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = (coda‘𝐺))
33	5	homacd 17943	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (coda‘𝐺) = 𝑍)
34	18, 33	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝐺) = 𝑍)
35	32, 34	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (coda‘𝑔) = 𝑍)
36	35	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (coda‘𝑔) = 𝑍)
37	21	adantrr 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (doma‘𝑔) = 𝑌)
38	31, 37	opeq12d 4828	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(doma‘𝑓), (doma‘𝑔)⟩ = ⟨𝑋, 𝑌⟩)
39	38, 36	oveq12d 7359	. . . . 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 6821	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑔) = (2nd ‘𝐺))
42	26	fveq2d 6821	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2nd ‘𝑓) = (2nd ‘𝐹))
43	39, 41, 42	oveq123d 7362	. . . 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 4835	. . 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 7499	. 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 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436  ⟨cop 4577  ⟨cotp 4579  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  2^nd c2nd 7915  compcco 17168  dom_acdoma 17922  cod_accoda 17923  Arrowcarw 17924  Hom_achoma 17925  comp_accoa 17956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-ot 4580  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-doma 17926  df-coda 17927  df-homa 17928  df-arw 17929  df-coa 17958

This theorem is referenced by:  coa2  17971  coahom  17972  arwlid  17974  arwrid  17975  arwass  17976