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Theorem prdsringd 20218

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 20141	. . . . 5 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Grp)
6	5	ssriv 3981	. . . 4 ⊢ Ring ⊆ Grp
7		fss 6727	. . . 4 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 585	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 18976	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		eqid 2726	. . . 4 ⊢ (𝑆X_s(mulGrp ∘ 𝑅)) = (𝑆X_s(mulGrp ∘ 𝑅))
11		mgpf 20151	. . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd
12		fco2 6737	. . . . 5 ⊢ (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
13	11, 4, 12	sylancr 586	. . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
14	10, 2, 3, 13	prdsmndd 18698	. . 3 ⊢ (𝜑 → (𝑆X_s(mulGrp ∘ 𝑅)) ∈ Mnd)
15		eqidd 2727	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16		eqid 2726	. . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌)
17	4	ffnd 6711	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
18	1, 16, 10, 2, 3, 17	prdsmgp 20054	. . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))) ∧ (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅)))))
19	18	simpld 494	. . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆X_s(mulGrp ∘ 𝑅))))
20	18	simprd 495	. . . . 5 ⊢ (𝜑 → (+_g‘(mulGrp‘𝑌)) = (+_g‘(𝑆X_s(mulGrp ∘ 𝑅))))
21	20	oveqdr 7432	. . . 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 18690	. . 3 ⊢ (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆X_s(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 7079	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑅‘𝑤) ∈ Ring)
26		eqid 2726	. . . . . . . . 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 1212	. . . . . . . . 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 17425	. . . . . . . 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 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 17425	. . . . . . . 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 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 17425	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤 ∈ 𝐼) → (𝑧‘𝑤) ∈ (Base‘(𝑅‘𝑤)))
42		eqid 2726	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑤)) = (Base‘(𝑅‘𝑤))
43		eqid 2726	. . . . . . . . 9 ⊢ (+_g‘(𝑅‘𝑤)) = (+_g‘(𝑅‘𝑤))
44		eqid 2726	. . . . . . . . 9 ⊢ (._r‘(𝑅‘𝑤)) = (._r‘(𝑅‘𝑤))
45	42, 43, 44	ringdi 20161	. . . . . . . 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 2726	. . . . . . . . 9 ⊢ (+_g‘𝑌) = (+_g‘𝑌)
48	1, 26, 28, 30, 32, 37, 40, 47, 34	prdsplusgfval 17427	. . . . . . . 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 7420	. . . . . . 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 2726	. . . . . . . . 9 ⊢ (._r‘𝑌) = (._r‘𝑌)
51	1, 26, 28, 30, 32, 33, 37, 50, 34	prdsmulrfval 17429	. . . . . . . 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 17429	. . . . . . . 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 7422	. . . . . . 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 2776	. . . . . 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 5241	. . . . 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 20146	. . . . . . . . . 10 ⊢ (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
58	57	ssriv 3981	. . . . . . . . 9 ⊢ Ring ⊆ Mnd
59		fss 6727	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
60	4, 58, 59	sylancl 585	. . . . . . . 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 18696	. . . . . 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 17428	. . . . 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 20217	. . . . . 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 20217	. . . . . 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 17426	. . . . 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 2776	. . . 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 20162	. . . . . . . 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 17427	. . . . . . . 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 7419	. . . . . . 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 17429	. . . . . . . 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 7422	. . . . . . 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 2776	. . . . . 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 5241	. . . . 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 18696	. . . . . 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 17428	. . . . 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 20217	. . . . . 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 17426	. . . . 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 2776	. . . 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 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 20140	. 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 ↦ cmpt 5224 ↾ cres 5671 ∘ ccom 5673 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7404 Basecbs 17151 +_gcplusg 17204 ._rcmulr 17205 X_scprds 17398 Mndcmnd 18665 Grpcgrp 18861 mulGrpcmgp 20037 Ringcrg 20136

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 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 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-prds 17400 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138

This theorem is referenced by: prdscrngd 20219 pwsring 20221 xpsringd 20229

W3C validator