Theorem prdsringd 19946

Description: A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)

Hypotheses

Ref	Expression
prdsringd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdsringd.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdsringd.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsringd.r	⊢ (𝜑 → 𝑅:𝐼⟶Ring)

Assertion

Ref	Expression
prdsringd	⊢ (𝜑 → 𝑌 ∈ Ring)

Proof of Theorem prdsringd

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsringd.y	. . 3 ⊢ 𝑌 = (𝑆Xs𝑅)
2		prdsringd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdsringd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdsringd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Ring)
5		ringgrp 19883	. . . . 5 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Grp)
6	5	ssriv 3940	. . . 4 ⊢ Ring ⊆ Grp
7		fss 6673	. . . 4 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 587	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 18782	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		eqid 2737	. . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11		mgpf 19893	. . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd
12		fco2 6683	. . . . 5 ⊢ (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
13	11, 4, 12	sylancr 588	. . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
14	10, 2, 3, 13	prdsmndd 18516	. . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15		eqidd 2738	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16		eqid 2737	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
17	4	ffnd 6657	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
18	1, 16, 10, 2, 3, 17	prdsmgp 19944	. . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
19	18	simpld 496	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
20	18	simprd 497	. . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
21	20	oveqdr 7370	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
22	15, 19, 21	mndpropd 18508	. . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
23	14, 22	mpbird 257	. 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
24	4	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
25	24	ffvelcdmda 7022	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑅‘𝑤) ∈ Ring)
26		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	3	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆 ∈ 𝑉)
28	27	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑆 ∈ 𝑉)
29	2	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼 ∈ 𝑊)
30	29	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	17	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
32	31	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑅 Fn 𝐼)
33		simplr1 1215	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑥 ∈ (Base‘𝑌))
34		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑤 ∈ 𝐼)
35	1, 26, 28, 30, 32, 33, 34	prdsbasprj 17281	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑥‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
36		simpr2 1195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
37	36	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑦 ∈ (Base‘𝑌))
38	1, 26, 28, 30, 32, 37, 34	prdsbasprj 17281	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑦‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
39		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
40	39	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑧 ∈ (Base‘𝑌))
41	1, 26, 28, 30, 32, 40, 34	prdsbasprj 17281	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
42		eqid 2737	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑤)) = (Base‘(𝑅‘𝑤))
43		eqid 2737	. . . . . . . . 9 ⊢ (+g‘(𝑅‘𝑤)) = (+g‘(𝑅‘𝑤))
44		eqid 2737	. . . . . . . . 9 ⊢ (.r‘(𝑅‘𝑤)) = (.r‘(𝑅‘𝑤))
45	42, 43, 44	ringdi 19900	. . . . . . . 8 ⊢ (((𝑅‘𝑤) ∈ Ring ∧ ((𝑥‘𝑤) ∈ (Base‘(𝑅‘𝑤)) ∧ (𝑦‘𝑤) ∈ (Base‘(𝑅‘𝑤)) ∧ (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))) → ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦‘𝑤)(+g‘(𝑅‘𝑤))(𝑧‘𝑤))) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑦‘𝑤))(+g‘(𝑅‘𝑤))((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
46	25, 35, 38, 41, 45	syl13anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦‘𝑤)(+g‘(𝑅‘𝑤))(𝑧‘𝑤))) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑦‘𝑤))(+g‘(𝑅‘𝑤))((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
47		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑌) = (+g‘𝑌)
48	1, 26, 28, 30, 32, 37, 40, 47, 34	prdsplusgfval 17283	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑦(+g‘𝑌)𝑧)‘𝑤) = ((𝑦‘𝑤)(+g‘(𝑅‘𝑤))(𝑧‘𝑤)))
49	48	oveq2d 7358	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦(+g‘𝑌)𝑧)‘𝑤)) = ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦‘𝑤)(+g‘(𝑅‘𝑤))(𝑧‘𝑤))))
50		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑌) = (.r‘𝑌)
51	1, 26, 28, 30, 32, 33, 37, 50, 34	prdsmulrfval 17285	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥(.r‘𝑌)𝑦)‘𝑤) = ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑦‘𝑤)))
52	1, 26, 28, 30, 32, 33, 40, 50, 34	prdsmulrfval 17285	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥(.r‘𝑌)𝑧)‘𝑤) = ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤)))
53	51, 52	oveq12d 7360	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (((𝑥(.r‘𝑌)𝑦)‘𝑤)(+g‘(𝑅‘𝑤))((𝑥(.r‘𝑌)𝑧)‘𝑤)) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑦‘𝑤))(+g‘(𝑅‘𝑤))((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
54	46, 49, 53	3eqtr4d 2787	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦(+g‘𝑌)𝑧)‘𝑤)) = (((𝑥(.r‘𝑌)𝑦)‘𝑤)(+g‘(𝑅‘𝑤))((𝑥(.r‘𝑌)𝑧)‘𝑤)))
55	54	mpteq2dva 5197	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑤 ∈ 𝐼 ↦ ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦(+g‘𝑌)𝑧)‘𝑤))) = (𝑤 ∈ 𝐼 ↦ (((𝑥(.r‘𝑌)𝑦)‘𝑤)(+g‘(𝑅‘𝑤))((𝑥(.r‘𝑌)𝑧)‘𝑤))))
56		simpr1 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
57		ringmnd 19888	. . . . . . . . . 10 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
58	57	ssriv 3940	. . . . . . . . 9 ⊢ Ring ⊆ Mnd
59		fss 6673	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
60	4, 58, 59	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
61	60	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
62	1, 26, 47, 27, 29, 61, 36, 39	prdsplusgcl 18514	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g‘𝑌)𝑧) ∈ (Base‘𝑌))
63	1, 26, 27, 29, 31, 56, 62, 50	prdsmulrval 17284	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r‘𝑌)(𝑦(+g‘𝑌)𝑧)) = (𝑤 ∈ 𝐼 ↦ ((𝑥‘𝑤)(.r‘(𝑅‘𝑤))((𝑦(+g‘𝑌)𝑧)‘𝑤))))
64	1, 26, 50, 27, 29, 24, 56, 36	prdsmulrcl 19945	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r‘𝑌)𝑦) ∈ (Base‘𝑌))
65	1, 26, 50, 27, 29, 24, 56, 39	prdsmulrcl 19945	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r‘𝑌)𝑧) ∈ (Base‘𝑌))
66	1, 26, 27, 29, 31, 64, 65, 47	prdsplusgval 17282	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r‘𝑌)𝑦)(+g‘𝑌)(𝑥(.r‘𝑌)𝑧)) = (𝑤 ∈ 𝐼 ↦ (((𝑥(.r‘𝑌)𝑦)‘𝑤)(+g‘(𝑅‘𝑤))((𝑥(.r‘𝑌)𝑧)‘𝑤))))
67	55, 63, 66	3eqtr4d 2787	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r‘𝑌)(𝑦(+g‘𝑌)𝑧)) = ((𝑥(.r‘𝑌)𝑦)(+g‘𝑌)(𝑥(.r‘𝑌)𝑧)))
68	42, 43, 44	ringdir 19901	. . . . . . . 8 ⊢ (((𝑅‘𝑤) ∈ Ring ∧ ((𝑥‘𝑤) ∈ (Base‘(𝑅‘𝑤)) ∧ (𝑦‘𝑤) ∈ (Base‘(𝑅‘𝑤)) ∧ (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))) → (((𝑥‘𝑤)(+g‘(𝑅‘𝑤))(𝑦‘𝑤))(.r‘(𝑅‘𝑤))(𝑧‘𝑤)) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))(+g‘(𝑅‘𝑤))((𝑦‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
69	25, 35, 38, 41, 68	syl13anc 1372	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (((𝑥‘𝑤)(+g‘(𝑅‘𝑤))(𝑦‘𝑤))(.r‘(𝑅‘𝑤))(𝑧‘𝑤)) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))(+g‘(𝑅‘𝑤))((𝑦‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
70	1, 26, 28, 30, 32, 33, 37, 47, 34	prdsplusgfval 17283	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑥(+g‘𝑌)𝑦)‘𝑤) = ((𝑥‘𝑤)(+g‘(𝑅‘𝑤))(𝑦‘𝑤)))
71	70	oveq1d 7357	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (((𝑥(+g‘𝑌)𝑦)‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤)) = (((𝑥‘𝑤)(+g‘(𝑅‘𝑤))(𝑦‘𝑤))(.r‘(𝑅‘𝑤))(𝑧‘𝑤)))
72	1, 26, 28, 30, 32, 37, 40, 50, 34	prdsmulrfval 17285	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → ((𝑦(.r‘𝑌)𝑧)‘𝑤) = ((𝑦‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤)))
73	52, 72	oveq12d 7360	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (((𝑥(.r‘𝑌)𝑧)‘𝑤)(+g‘(𝑅‘𝑤))((𝑦(.r‘𝑌)𝑧)‘𝑤)) = (((𝑥‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))(+g‘(𝑅‘𝑤))((𝑦‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
74	69, 71, 73	3eqtr4d 2787	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (((𝑥(+g‘𝑌)𝑦)‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤)) = (((𝑥(.r‘𝑌)𝑧)‘𝑤)(+g‘(𝑅‘𝑤))((𝑦(.r‘𝑌)𝑧)‘𝑤)))
75	74	mpteq2dva 5197	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑤 ∈ 𝐼 ↦ (((𝑥(+g‘𝑌)𝑦)‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))) = (𝑤 ∈ 𝐼 ↦ (((𝑥(.r‘𝑌)𝑧)‘𝑤)(+g‘(𝑅‘𝑤))((𝑦(.r‘𝑌)𝑧)‘𝑤))))
76	1, 26, 47, 27, 29, 61, 56, 36	prdsplusgcl 18514	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g‘𝑌)𝑦) ∈ (Base‘𝑌))
77	1, 26, 27, 29, 31, 76, 39, 50	prdsmulrval 17284	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g‘𝑌)𝑦)(.r‘𝑌)𝑧) = (𝑤 ∈ 𝐼 ↦ (((𝑥(+g‘𝑌)𝑦)‘𝑤)(.r‘(𝑅‘𝑤))(𝑧‘𝑤))))
78	1, 26, 50, 27, 29, 24, 36, 39	prdsmulrcl 19945	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r‘𝑌)𝑧) ∈ (Base‘𝑌))
79	1, 26, 27, 29, 31, 65, 78, 47	prdsplusgval 17282	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r‘𝑌)𝑧)(+g‘𝑌)(𝑦(.r‘𝑌)𝑧)) = (𝑤 ∈ 𝐼 ↦ (((𝑥(.r‘𝑌)𝑧)‘𝑤)(+g‘(𝑅‘𝑤))((𝑦(.r‘𝑌)𝑧)‘𝑤))))
80	75, 77, 79	3eqtr4d 2787	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g‘𝑌)𝑦)(.r‘𝑌)𝑧) = ((𝑥(.r‘𝑌)𝑧)(+g‘𝑌)(𝑦(.r‘𝑌)𝑧)))
81	67, 80	jca 513	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r‘𝑌)(𝑦(+g‘𝑌)𝑧)) = ((𝑥(.r‘𝑌)𝑦)(+g‘𝑌)(𝑥(.r‘𝑌)𝑧)) ∧ ((𝑥(+g‘𝑌)𝑦)(.r‘𝑌)𝑧) = ((𝑥(.r‘𝑌)𝑧)(+g‘𝑌)(𝑦(.r‘𝑌)𝑧))))
82	81	ralrimivvva 3197	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)∀𝑧 ∈ (Base‘𝑌)((𝑥(.r‘𝑌)(𝑦(+g‘𝑌)𝑧)) = ((𝑥(.r‘𝑌)𝑦)(+g‘𝑌)(𝑥(.r‘𝑌)𝑧)) ∧ ((𝑥(+g‘𝑌)𝑦)(.r‘𝑌)𝑧) = ((𝑥(.r‘𝑌)𝑧)(+g‘𝑌)(𝑦(.r‘𝑌)𝑧))))
83	26, 16, 47, 50	isring 19882	. 2 ⊢ (𝑌 ∈ Ring ↔ (𝑌 ∈ Grp ∧ (mulGrp‘𝑌) ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)∀𝑧 ∈ (Base‘𝑌)((𝑥(.r‘𝑌)(𝑦(+g‘𝑌)𝑧)) = ((𝑥(.r‘𝑌)𝑦)(+g‘𝑌)(𝑥(.r‘𝑌)𝑧)) ∧ ((𝑥(+g‘𝑌)𝑦)(.r‘𝑌)𝑧) = ((𝑥(.r‘𝑌)𝑧)(+g‘𝑌)(𝑦(.r‘𝑌)𝑧)))))
84	9, 23, 82, 83	syl3anbrc 1343	1 ⊢ (𝜑 → 𝑌 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3062   ⊆ wss 3902   ↦ cmpt 5180   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6479  ⟶wf 6480  ‘cfv 6484  (class class class)co 7342  Basecbs 17010  +_gcplusg 17060  ._rcmulr 17061  X_scprds 17254  Mndcmnd 18483  Grpcgrp 18674  mulGrpcmgp 19815  Ringcrg 19878

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 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-er 8574  df-map 8693  df-ixp 8762  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-sup 9304  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-2 12142  df-3 12143  df-4 12144  df-5 12145  df-6 12146  df-7 12147  df-8 12148  df-9 12149  df-n0 12340  df-z 12426  df-dec 12544  df-uz 12689  df-fz 13346  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-plusg 17073  df-mulr 17074  df-sca 17076  df-vsca 17077  df-ip 17078  df-tset 17079  df-ple 17080  df-ds 17082  df-hom 17084  df-cco 17085  df-0g 17250  df-prds 17256  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-grp 18677  df-minusg 18678  df-mgp 19816  df-ring 19880

This theorem is referenced by:  prdscrngd  19947  pwsring  19949