Theorem prdsringd 20225

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 20142	. . . . 5 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Grp)
6	5	ssriv 3941	. . . 4 ⊢ Ring ⊆ Grp
7		fss 6672	. . . 4 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 18948	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		eqid 2729	. . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11		mgpf 20152	. . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd
12		fco2 6682	. . . . 5 ⊢ (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
13	11, 4, 12	sylancr 587	. . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
14	10, 2, 3, 13	prdsmndd 18663	. . 3 ⊢ (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15		eqidd 2730	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16		eqid 2729	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
17	4	ffnd 6657	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
18	1, 16, 10, 2, 3, 17	prdsmgp 20055	. . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
19	18	simpld 494	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
20	18	simprd 495	. . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
21	20	oveqdr 7381	. . . 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 18652	. . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
23	14, 22	mpbird 257	. 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
24	4	adantr 480	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆 ∈ 𝑉)
28	27	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑆 ∈ 𝑉)
29	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼 ∈ 𝑊)
30	29	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
32	31	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑅 Fn 𝐼)
33		simplr1 1216	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑥 ∈ (Base‘𝑌))
34		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑤 ∈ 𝐼)
35	1, 26, 28, 30, 32, 33, 34	prdsbasprj 17395	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑥‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
36		simpr2 1196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
37	36	adantr 480	. . . . . . . . 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 17395	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑦‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
39		simpr3 1197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
40	39	adantr 480	. . . . . . . . 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 17395	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
42		eqid 2729	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑤)) = (Base‘(𝑅‘𝑤))
43		eqid 2729	. . . . . . . . 9 ⊢ (+g‘(𝑅‘𝑤)) = (+g‘(𝑅‘𝑤))
44		eqid 2729	. . . . . . . . 9 ⊢ (.r‘(𝑅‘𝑤)) = (.r‘(𝑅‘𝑤))
45	42, 43, 44	ringdi 20165	. . . . . . . 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 1374	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (+g‘𝑌) = (+g‘𝑌)
48	1, 26, 28, 30, 32, 37, 40, 47, 34	prdsplusgfval 17397	. . . . . . . 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 7369	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (.r‘𝑌) = (.r‘𝑌)
51	1, 26, 28, 30, 32, 33, 37, 50, 34	prdsmulrfval 17399	. . . . . . . 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 17399	. . . . . . . 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 7371	. . . . . . 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 2774	. . . . . 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 5188	. . . . 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 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
57		ringmnd 20147	. . . . . . . . . 10 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
58	57	ssriv 3941	. . . . . . . . 9 ⊢ Ring ⊆ Mnd
59		fss 6672	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
60	4, 58, 59	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
61	60	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
62	1, 26, 47, 27, 29, 61, 36, 39	prdsplusgcl 18661	. . . . . 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 17398	. . . . 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 20224	. . . . . 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 20224	. . . . . 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 17396	. . . . 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 2774	. . . 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 20166	. . . . . . . 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 1374	. . . . . . 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 17397	. . . . . . . 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 7368	. . . . . . 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 17399	. . . . . . . 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 7371	. . . . . . 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 2774	. . . . . 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 5188	. . . . 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 18661	. . . . . 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 17398	. . . . 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 20224	. . . . . 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 17396	. . . . 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 2774	. . . 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 511	. . 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 3175	. 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 20141	. 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 1344	1 ⊢ (𝜑 → 𝑌 ∈ Ring)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3905   ↦ cmpt 5176   ↾ cres 5625   ∘ ccom 5627   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  +_gcplusg 17180  ._rcmulr 17181  X_scprds 17368  Mndcmnd 18627  Grpcgrp 18831  mulGrpcmgp 20044  Ringcrg 20137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-map 8762  df-ixp 8832  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12611  df-uz 12755  df-fz 13430  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-plusg 17193  df-mulr 17194  df-sca 17196  df-vsca 17197  df-ip 17198  df-tset 17199  df-ple 17200  df-ds 17202  df-hom 17204  df-cco 17205  df-0g 17364  df-prds 17370  df-mgm 18533  df-sgrp 18612  df-mnd 18628  df-grp 18834  df-minusg 18835  df-cmn 19680  df-abl 19681  df-mgp 20045  df-rng 20057  df-ur 20086  df-ring 20139

This theorem is referenced by:  prdscrngd  20226  pwsring  20228  xpsringd  20236