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Theorem coaval 18057

Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
homdmcoa.o	⊢ · = (comp_a‘𝐶)
homdmcoa.h	⊢ 𝐻 = (Hom_a‘𝐶)
homdmcoa.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
coaval.x	⊢ ∙ = (comp‘𝐶)

**Assertion**
Ref	Expression
coaval	⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2^nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2^nd ‘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 ⊢ · = (comp_a‘𝐶)
2		eqid 2728	. . 3 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
3		coaval.x	. . 3 ⊢ ∙ = (comp‘𝐶)
4	1, 2, 3	coafval 18053	. 2 ⊢ · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩)
5		homdmcoa.h	. . . . 5 ⊢ 𝐻 = (Hom_a‘𝐶)
6	2, 5	homarw 18035	. . . 4 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3978	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9		fveqeq2 6906	. . . 4 ⊢ (ℎ = 𝐹 → ((cod_a‘ℎ) = (dom_a‘𝑔) ↔ (cod_a‘𝐹) = (dom_a‘𝑔)))
10	2, 5	homarw 18035	. . . . 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 3978	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
14	5	homacd 18030	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)
15	12, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝐹) = 𝑌)
16		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
17	16	fveq2d 6901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝑔) = (dom_a‘𝐺))
18	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
19	5	homadm 18029	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (dom_a‘𝐺) = 𝑌)
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝐺) = 𝑌)
21	17, 20	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝑔) = 𝑌)
22	15, 21	eqtr4d 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝐹) = (dom_a‘𝑔))
23	9, 13, 22	elrabd 3684	. . 3 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)})
24		otex 5467	. . . 4 ⊢ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩ ∈ V
25	24	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩ ∈ V)
26		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
27	26	fveq2d 6901	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑓) = (dom_a‘𝐹))
28	5	homadm 18029	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)
29	12, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝐹) = 𝑋)
30	29	adantrr 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝐹) = 𝑋)
31	27, 30	eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑓) = 𝑋)
32	16	fveq2d 6901	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝑔) = (cod_a‘𝐺))
33	5	homacd 18030	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (cod_a‘𝐺) = 𝑍)
34	18, 33	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝐺) = 𝑍)
35	32, 34	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝑔) = 𝑍)
36	35	adantrr 716	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (cod_a‘𝑔) = 𝑍)
37	21	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑔) = 𝑌)
38	31, 37	opeq12d 4882	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ = ⟨𝑋, 𝑌⟩)
39	38, 36	oveq12d 7438	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔)) = (⟨𝑋, 𝑌⟩ ∙ 𝑍))
40		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
41	40	fveq2d 6901	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑔) = (2^nd ‘𝐺))
42	26	fveq2d 6901	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑓) = (2^nd ‘𝐹))
43	39, 41, 42	oveq123d 7441	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓)) = ((2^nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2^nd ‘𝐹)))
44	31, 36, 43	oteq123d 4889	. . 3 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩ = ⟨𝑋, 𝑍, ((2^nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2^nd ‘𝐹))⟩)
45	8, 23, 25, 44	ovmpodv2 7579	. 2 ⊢ (𝜑 → ( · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ {ℎ ∈ (Arrow‘𝐶) ∣ (cod_a‘ℎ) = (dom_a‘𝑔)} ↦ ⟨(dom_a‘𝑓), (cod_a‘𝑔), ((2^nd ‘𝑔)(⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔))(2^nd ‘𝑓))⟩) → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2^nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2^nd ‘𝐹))⟩))
46	4, 45	mpi 20	1 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2^nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2^nd ‘𝐹))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 ⟨cop 4635 ⟨cotp 4637 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 2^nd c2nd 7992 compcco 17245 dom_acdoma 18009 cod_accoda 18010 Arrowcarw 18011 Hom_achoma 18012 comp_accoa 18043

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

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 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-ot 4638 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-doma 18013 df-coda 18014 df-homa 18015 df-arw 18016 df-coa 18045

This theorem is referenced by: coa2 18058 coahom 18059 arwlid 18061 arwrid 18062 arwass 18063

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