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Theorem coaval 18014

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 2732	. . 3 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
3		coaval.x	. . 3 ⊢ ∙ = (comp‘𝐶)
4	1, 2, 3	coafval 18010	. 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 17992	. . . 4 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3979	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9		fveqeq2 6897	. . . 4 ⊢ (ℎ = 𝐹 → ((cod_a‘ℎ) = (dom_a‘𝑔) ↔ (cod_a‘𝐹) = (dom_a‘𝑔)))
10	2, 5	homarw 17992	. . . . 5 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
11		homdmcoa.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
12	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
13	10, 12	sselid 3979	. . . 4 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
14	5	homacd 17987	. . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (cod_a‘𝐹) = 𝑌)
15	12, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝐹) = 𝑌)
16		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
17	16	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝑔) = (dom_a‘𝐺))
18	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
19	5	homadm 17986	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (dom_a‘𝐺) = 𝑌)
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝐺) = 𝑌)
21	17, 20	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝑔) = 𝑌)
22	15, 21	eqtr4d 2775	. . . 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 5464	. . . 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 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑓 = 𝐹)
27	26	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑓) = (dom_a‘𝐹))
28	5	homadm 17986	. . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)
29	12, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (dom_a‘𝐹) = 𝑋)
30	29	adantrr 715	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝐹) = 𝑋)
31	27, 30	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑓) = 𝑋)
32	16	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝑔) = (cod_a‘𝐺))
33	5	homacd 17987	. . . . . . 7 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (cod_a‘𝐺) = 𝑍)
34	18, 33	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝐺) = 𝑍)
35	32, 34	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝐺) → (cod_a‘𝑔) = 𝑍)
36	35	adantrr 715	. . . 4 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (cod_a‘𝑔) = 𝑍)
37	21	adantrr 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (dom_a‘𝑔) = 𝑌)
38	31, 37	opeq12d 4880	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → ⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ = ⟨𝑋, 𝑌⟩)
39	38, 36	oveq12d 7423	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (⟨(dom_a‘𝑓), (dom_a‘𝑔)⟩ ∙ (cod_a‘𝑔)) = (⟨𝑋, 𝑌⟩ ∙ 𝑍))
40		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → 𝑔 = 𝐺)
41	40	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑔) = (2^nd ‘𝐺))
42	26	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (2^nd ‘𝑓) = (2^nd ‘𝐹))
43	39, 41, 42	oveq123d 7426	. . . 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 4887	. . 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 7562	. 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 396 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ⟨cop 4633 ⟨cotp 4635 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 2^nd c2nd 7970 compcco 17205 dom_acdoma 17966 cod_accoda 17967 Arrowcarw 17968 Hom_achoma 17969 comp_accoa 18000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-doma 17970 df-coda 17971 df-homa 17972 df-arw 17973 df-coa 18002

This theorem is referenced by: coa2 18015 coahom 18016 arwlid 18018 arwrid 18019 arwass 18020

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