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Theorem prdsringd 20286
Description: A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
Hypotheses
Ref Expression
prdsringd.y 𝑌 = (𝑆Xs𝑅)
prdsringd.i (𝜑𝐼𝑊)
prdsringd.s (𝜑𝑆𝑉)
prdsringd.r (𝜑𝑅:𝐼⟶Ring)
Assertion
Ref Expression
prdsringd (𝜑𝑌 ∈ Ring)

Proof of Theorem prdsringd
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsringd.y . . 3 𝑌 = (𝑆Xs𝑅)
2 prdsringd.i . . 3 (𝜑𝐼𝑊)
3 prdsringd.s . . 3 (𝜑𝑆𝑉)
4 prdsringd.r . . . 4 (𝜑𝑅:𝐼⟶Ring)
5 ringgrp 20207 . . . . 5 (𝑥 ∈ Ring → 𝑥 ∈ Grp)
65ssriv 3980 . . . 4 Ring ⊆ Grp
7 fss 6739 . . . 4 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 584 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 19030 . 2 (𝜑𝑌 ∈ Grp)
10 eqid 2725 . . . 4 (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11 mgpf 20217 . . . . 5 (mulGrp ↾ Ring):Ring⟶Mnd
12 fco2 6750 . . . . 5 (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1311, 4, 12sylancr 585 . . . 4 (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1410, 2, 3, 13prdsmndd 18746 . . 3 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15 eqidd 2726 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16 eqid 2725 . . . . . 6 (mulGrp‘𝑌) = (mulGrp‘𝑌)
174ffnd 6724 . . . . . 6 (𝜑𝑅 Fn 𝐼)
181, 16, 10, 2, 3, 17prdsmgp 20120 . . . . 5 (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
1918simpld 493 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
2018simprd 494 . . . . 5 (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
2120oveqdr 7447 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
2215, 19, 21mndpropd 18738 . . 3 (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
2314, 22mpbird 256 . 2 (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
244adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
2524ffvelcdmda 7093 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑅𝑤) ∈ Ring)
26 eqid 2725 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
273adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆𝑉)
2827adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑆𝑉)
292adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼𝑊)
3029adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝐼𝑊)
3117adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
3231adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑅 Fn 𝐼)
33 simplr1 1212 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑥 ∈ (Base‘𝑌))
34 simpr 483 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑤𝐼)
351, 26, 28, 30, 32, 33, 34prdsbasprj 17473 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑥𝑤) ∈ (Base‘(𝑅𝑤)))
36 simpr2 1192 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
3736adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑦 ∈ (Base‘𝑌))
381, 26, 28, 30, 32, 37, 34prdsbasprj 17473 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑦𝑤) ∈ (Base‘(𝑅𝑤)))
39 simpr3 1193 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
4039adantr 479 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑧 ∈ (Base‘𝑌))
411, 26, 28, 30, 32, 40, 34prdsbasprj 17473 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))
42 eqid 2725 . . . . . . . . 9 (Base‘(𝑅𝑤)) = (Base‘(𝑅𝑤))
43 eqid 2725 . . . . . . . . 9 (+g‘(𝑅𝑤)) = (+g‘(𝑅𝑤))
44 eqid 2725 . . . . . . . . 9 (.r‘(𝑅𝑤)) = (.r‘(𝑅𝑤))
4542, 43, 44ringdi 20229 . . . . . . . 8 (((𝑅𝑤) ∈ Ring ∧ ((𝑥𝑤) ∈ (Base‘(𝑅𝑤)) ∧ (𝑦𝑤) ∈ (Base‘(𝑅𝑤)) ∧ (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))) → ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤))) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤))(+g‘(𝑅𝑤))((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
4625, 35, 38, 41, 45syl13anc 1369 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤))) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤))(+g‘(𝑅𝑤))((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
47 eqid 2725 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
481, 26, 28, 30, 32, 37, 40, 47, 34prdsplusgfval 17475 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(+g𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤)))
4948oveq2d 7435 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤)) = ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤))))
50 eqid 2725 . . . . . . . . 9 (.r𝑌) = (.r𝑌)
511, 26, 28, 30, 32, 33, 37, 50, 34prdsmulrfval 17477 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤)))
521, 26, 28, 30, 32, 33, 40, 50, 34prdsmulrfval 17477 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑧)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
5351, 52oveq12d 7437 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (((𝑥(.r𝑌)𝑦)‘𝑤)(+g‘(𝑅𝑤))((𝑥(.r𝑌)𝑧)‘𝑤)) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤))(+g‘(𝑅𝑤))((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
5446, 49, 533eqtr4d 2775 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤)) = (((𝑥(.r𝑌)𝑦)‘𝑤)(+g‘(𝑅𝑤))((𝑥(.r𝑌)𝑧)‘𝑤)))
5554mpteq2dva 5249 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑤𝐼 ↦ ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤))) = (𝑤𝐼 ↦ (((𝑥(.r𝑌)𝑦)‘𝑤)(+g‘(𝑅𝑤))((𝑥(.r𝑌)𝑧)‘𝑤))))
56 simpr1 1191 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
57 ringmnd 20212 . . . . . . . . . 10 (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
5857ssriv 3980 . . . . . . . . 9 Ring ⊆ Mnd
59 fss 6739 . . . . . . . . 9 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
604, 58, 59sylancl 584 . . . . . . . 8 (𝜑𝑅:𝐼⟶Mnd)
6160adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
621, 26, 47, 27, 29, 61, 36, 39prdsplusgcl 18744 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g𝑌)𝑧) ∈ (Base‘𝑌))
631, 26, 27, 29, 31, 56, 62, 50prdsmulrval 17476 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = (𝑤𝐼 ↦ ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤))))
641, 26, 50, 27, 29, 24, 56, 36prdsmulrcl 20285 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑦) ∈ (Base‘𝑌))
651, 26, 50, 27, 29, 24, 56, 39prdsmulrcl 20285 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑧) ∈ (Base‘𝑌))
661, 26, 27, 29, 31, 64, 65, 47prdsplusgval 17474 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)) = (𝑤𝐼 ↦ (((𝑥(.r𝑌)𝑦)‘𝑤)(+g‘(𝑅𝑤))((𝑥(.r𝑌)𝑧)‘𝑤))))
6755, 63, 663eqtr4d 2775 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)))
6842, 43, 44ringdir 20230 . . . . . . . 8 (((𝑅𝑤) ∈ Ring ∧ ((𝑥𝑤) ∈ (Base‘(𝑅𝑤)) ∧ (𝑦𝑤) ∈ (Base‘(𝑅𝑤)) ∧ (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))) → (((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤))(.r‘(𝑅𝑤))(𝑧𝑤)) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))(+g‘(𝑅𝑤))((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
6925, 35, 38, 41, 68syl13anc 1369 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤))(.r‘(𝑅𝑤))(𝑧𝑤)) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))(+g‘(𝑅𝑤))((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
701, 26, 28, 30, 32, 33, 37, 47, 34prdsplusgfval 17475 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(+g𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤)))
7170oveq1d 7434 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)) = (((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤))(.r‘(𝑅𝑤))(𝑧𝑤)))
721, 26, 28, 30, 32, 37, 40, 50, 34prdsmulrfval 17477 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(.r𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
7352, 72oveq12d 7437 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (((𝑥(.r𝑌)𝑧)‘𝑤)(+g‘(𝑅𝑤))((𝑦(.r𝑌)𝑧)‘𝑤)) = (((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))(+g‘(𝑅𝑤))((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
7469, 71, 733eqtr4d 2775 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)) = (((𝑥(.r𝑌)𝑧)‘𝑤)(+g‘(𝑅𝑤))((𝑦(.r𝑌)𝑧)‘𝑤)))
7574mpteq2dva 5249 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑤𝐼 ↦ (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))) = (𝑤𝐼 ↦ (((𝑥(.r𝑌)𝑧)‘𝑤)(+g‘(𝑅𝑤))((𝑦(.r𝑌)𝑧)‘𝑤))))
761, 26, 47, 27, 29, 61, 56, 36prdsplusgcl 18744 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g𝑌)𝑦) ∈ (Base‘𝑌))
771, 26, 27, 29, 31, 76, 39, 50prdsmulrval 17476 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = (𝑤𝐼 ↦ (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
781, 26, 50, 27, 29, 24, 36, 39prdsmulrcl 20285 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r𝑌)𝑧) ∈ (Base‘𝑌))
791, 26, 27, 29, 31, 65, 78, 47prdsplusgval 17474 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧)) = (𝑤𝐼 ↦ (((𝑥(.r𝑌)𝑧)‘𝑤)(+g‘(𝑅𝑤))((𝑦(.r𝑌)𝑧)‘𝑤))))
8075, 77, 793eqtr4d 2775 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧)))
8167, 80jca 510 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)) ∧ ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧))))
8281ralrimivvva 3193 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)∀𝑧 ∈ (Base‘𝑌)((𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)) ∧ ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧))))
8326, 16, 47, 50isring 20206 . 2 (𝑌 ∈ Ring ↔ (𝑌 ∈ Grp ∧ (mulGrp‘𝑌) ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑌)∀𝑦 ∈ (Base‘𝑌)∀𝑧 ∈ (Base‘𝑌)((𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)) ∧ ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧)))))
849, 23, 82, 83syl3anbrc 1340 1 (𝜑𝑌 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wss 3944  cmpt 5232  cres 5680  ccom 5682   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  Basecbs 17199  +gcplusg 17252  .rcmulr 17253  Xscprds 17446  Mndcmnd 18713  Grpcgrp 18914  mulGrpcmgp 20103  Ringcrg 20202
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  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 17135  df-sets 17152  df-slot 17170  df-ndx 17182  df-base 17200  df-plusg 17265  df-mulr 17266  df-sca 17268  df-vsca 17269  df-ip 17270  df-tset 17271  df-ple 17272  df-ds 17274  df-hom 17276  df-cco 17277  df-0g 17442  df-prds 17448  df-mgm 18619  df-sgrp 18698  df-mnd 18714  df-grp 18917  df-minusg 18918  df-cmn 19766  df-abl 19767  df-mgp 20104  df-rng 20122  df-ur 20151  df-ring 20204
This theorem is referenced by:  prdscrngd  20287  pwsring  20289  xpsringd  20297
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