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Theorem prdsringd 20133

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 20060	. . . . 5 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Grp)
6	5	ssriv 3986	. . . 4 ⊢ Ring ⊆ Grp
7		fss 6734	. . . 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 18932	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		eqid 2732	. . . 4 ⊢ (𝑆X_s(mulGrp ∘ 𝑅)) = (𝑆X_s(mulGrp ∘ 𝑅))
11		mgpf 20070	. . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd
12		fco2 6744	. . . . 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 18657	. . 3 ⊢ (𝜑 → (𝑆X_s(mulGrp ∘ 𝑅)) ∈ Mnd)
15		eqidd 2733	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16		eqid 2732	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
17	4	ffnd 6718	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
18	1, 16, 10, 2, 3, 17	prdsmgp 20131	. . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))) ∧ (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅)))))
19	18	simpld 495	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))))
20	18	simprd 496	. . . . 5 ⊢ (𝜑 → (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅))))
21	20	oveqdr 7436	. . . 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 18649	. . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆X_s(mulGrp ∘ 𝑅)) ∈ Mnd))
23	14, 22	mpbird 256	. 2 ⊢ (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
24	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
25	24	ffvelcdmda 7086	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑅‘𝑤) ∈ Ring)
26		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝑌) = (Base‘𝑌)
27	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆 ∈ 𝑉)
28	27	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑆 ∈ 𝑉)
29	2	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼 ∈ 𝑊)
30	29	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝐼 ∈ 𝑊)
31	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
32	31	adantr 481	. . . . . . . . 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 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → 𝑤 ∈ 𝐼)
35	1, 26, 28, 30, 32, 33, 34	prdsbasprj 17417	. . . . . . . 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 481	. . . . . . . . 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 17417	. . . . . . . 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 481	. . . . . . . . 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 17417	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
42		eqid 2732	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑤)) = (Base‘(𝑅‘𝑤))
43		eqid 2732	. . . . . . . . 9 ⊢ (+_g‘(𝑅‘𝑤)) = (+_g‘(𝑅‘𝑤))
44		eqid 2732	. . . . . . . . 9 ⊢ (._r‘(𝑅‘𝑤)) = (._r‘(𝑅‘𝑤))
45	42, 43, 44	ringdi 20080	. . . . . . . 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 2732	. . . . . . . . 9 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
48	1, 26, 28, 30, 32, 37, 40, 47, 34	prdsplusgfval 17419	. . . . . . . 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 7424	. . . . . . 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 2732	. . . . . . . . 9 ⊢ (._r‘𝑌) = (._r‘𝑌)
51	1, 26, 28, 30, 32, 33, 37, 50, 34	prdsmulrfval 17421	. . . . . . . 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 17421	. . . . . . . 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 7426	. . . . . . 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 2782	. . . . . 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 5248	. . . . 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 20065	. . . . . . . . . 10 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
58	57	ssriv 3986	. . . . . . . . 9 ⊢ Ring ⊆ Mnd
59		fss 6734	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
60	4, 58, 59	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)
61	60	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
62	1, 26, 47, 27, 29, 61, 36, 39	prdsplusgcl 18655	. . . . . 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 17420	. . . . 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 20132	. . . . . 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 20132	. . . . . 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 17418	. . . . 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 2782	. . . 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 20081	. . . . . . . 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 17419	. . . . . . . 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 7423	. . . . . . 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 17421	. . . . . . . 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 7426	. . . . . . 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 2782	. . . . . 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 5248	. . . . 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 18655	. . . . . 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 17420	. . . . 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 20132	. . . . . 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 17418	. . . . 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 2782	. . . 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 512	. . 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 3203	. 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 20059	. 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)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3948 ↦ cmpt 5231 ↾ cres 5678 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 +_gcplusg 17196 ._rcmulr 17197 X_scprds 17390 Mndcmnd 18624 Grpcgrp 18818 mulGrpcmgp 19986 Ringcrg 20055

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-mgp 19987 df-ring 20057

This theorem is referenced by: prdscrngd 20134 pwsring 20136 xpsringd 20144

W3C validator