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Theorem prdsringd 20264

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

**Hypotheses**
Ref	Expression
prdsringd.y	⊢ 𝑌 = (𝑆X_s𝑅)
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 = (𝑆X_s𝑅)
2		prdsringd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdsringd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdsringd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Ring)
5		ringgrp 20185	. . . . 5 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Grp)
6	5	ssriv 3986	. . . 4 ⊢ Ring ⊆ Grp
7		fss 6744	. . . 4 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 584	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 19013	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		eqid 2728	. . . 4 ⊢ (𝑆X_s(mulGrp ∘ 𝑅)) = (𝑆X_s(mulGrp ∘ 𝑅))
11		mgpf 20195	. . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd
12		fco2 6755	. . . . 5 ⊢ (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
13	11, 4, 12	sylancr 585	. . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
14	10, 2, 3, 13	prdsmndd 18734	. . 3 ⊢ (𝜑 → (𝑆X_s(mulGrp ∘ 𝑅)) ∈ Mnd)
15		eqidd 2729	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16		eqid 2728	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
17	4	ffnd 6728	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
18	1, 16, 10, 2, 3, 17	prdsmgp 20098	. . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))) ∧ (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅)))))
19	18	simpld 493	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))))
20	18	simprd 494	. . . . 5 ⊢ (𝜑 → (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅))))
21	20	oveqdr 7454	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+_g‘(mulGrp‘𝑌))𝑦) = (𝑥(+_g‘(𝑆X_s(mulGrp ∘ 𝑅)))𝑦))
22	15, 19, 21	mndpropd 18726	. . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆X_s(mulGrp ∘ 𝑅)) ∈ Mnd))
23	14, 22	mpbird 256	. 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
24	4	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
25	24	ffvelcdmda 7099	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑅‘𝑤) ∈ Ring)
26		eqid 2728	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	3	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆 ∈ 𝑉)
28	27	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑆 ∈ 𝑉)
29	2	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼 ∈ 𝑊)
30	29	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	17	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
32	31	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑅 Fn 𝐼)
33		simplr1 1212	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑥 ∈ (Base‘𝑌))
34		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑤 ∈ 𝐼)
35	1, 26, 28, 30, 32, 33, 34	prdsbasprj 17461	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑥‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
36		simpr2 1192	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
37	36	adantr 479	. . . . . . . . 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 17461	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑦‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
39		simpr3 1193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
40	39	adantr 479	. . . . . . . . 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 17461	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
42		eqid 2728	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑤)) = (Base‘(𝑅‘𝑤))
43		eqid 2728	. . . . . . . . 9 ⊢ (+_g‘(𝑅‘𝑤)) = (+_g‘(𝑅‘𝑤))
44		eqid 2728	. . . . . . . . 9 ⊢ (._r‘(𝑅‘𝑤)) = (._r‘(𝑅‘𝑤))
45	42, 43, 44	ringdi 20207	. . . . . . . 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 1369	. . . . . . 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 2728	. . . . . . . . 9 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
48	1, 26, 28, 30, 32, 37, 40, 47, 34	prdsplusgfval 17463	. . . . . . . 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 7442	. . . . . . 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 2728	. . . . . . . . 9 ⊢ (._r‘𝑌) = (._r‘𝑌)
51	1, 26, 28, 30, 32, 33, 37, 50, 34	prdsmulrfval 17465	. . . . . . . 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 17465	. . . . . . . 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 7444	. . . . . . 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 2778	. . . . . 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 5252	. . . . 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 1191	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
57		ringmnd 20190	. . . . . . . . . 10 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
58	57	ssriv 3986	. . . . . . . . 9 ⊢ Ring ⊆ Mnd
59		fss 6744	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
60	4, 58, 59	sylancl 584	. . . . . . . 8 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
61	60	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
62	1, 26, 47, 27, 29, 61, 36, 39	prdsplusgcl 18732	. . . . . 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 17464	. . . . 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 20263	. . . . . 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 20263	. . . . . 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 17462	. . . . 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 2778	. . . 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 20208	. . . . . . . 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 1369	. . . . . . 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 17463	. . . . . . . 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 7441	. . . . . . 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 17465	. . . . . . . 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 7444	. . . . . . 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 2778	. . . . . 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 5252	. . . . 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 18732	. . . . . 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 17464	. . . . 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 20263	. . . . . 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 17462	. . . . 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 2778	. . . 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 510	. . 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 3201	. 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 20184	. 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 1340	1 ⊢ (𝜑 → 𝑌 ∈ Ring)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ⊆ wss 3949 ↦ cmpt 5235 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +_gcplusg 17240 ._rcmulr 17241 X_scprds 17434 Mndcmnd 18701 Grpcgrp 18897 mulGrpcmgp 20081 Ringcrg 20180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-prds 17436 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182

This theorem is referenced by: prdscrngd 20265 pwsring 20267 xpsringd 20275

W3C validator