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Theorem prdsringd 18658
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 18598 . . . . 5 (𝑥 ∈ Ring → 𝑥 ∈ Grp)
65ssriv 3640 . . . 4 Ring ⊆ Grp
7 fss 6094 . . . 4 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 695 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17572 . 2 (𝜑𝑌 ∈ Grp)
10 eqid 2651 . . . 4 (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11 mgpf 18605 . . . . 5 (mulGrp ↾ Ring):Ring⟶Mnd
12 fco2 6097 . . . . 5 (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1311, 4, 12sylancr 696 . . . 4 (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1410, 2, 3, 13prdsmndd 17370 . . 3 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15 eqidd 2652 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16 eqid 2651 . . . . . 6 (mulGrp‘𝑌) = (mulGrp‘𝑌)
17 ffn 6083 . . . . . . 7 (𝑅:𝐼⟶Ring → 𝑅 Fn 𝐼)
184, 17syl 17 . . . . . 6 (𝜑𝑅 Fn 𝐼)
191, 16, 10, 2, 3, 18prdsmgp 18656 . . . . 5 (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
2019simpld 474 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
2119simprd 478 . . . . 5 (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
2221oveqdr 6714 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
2315, 20, 22mndpropd 17363 . . 3 (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
2414, 23mpbird 247 . 2 (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
254adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
2625ffvelrnda 6399 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑅𝑤) ∈ Ring)
27 eqid 2651 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
283adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆𝑉)
2928adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑆𝑉)
302adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼𝑊)
3130adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝐼𝑊)
3218adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
3332adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑅 Fn 𝐼)
34 simplr1 1123 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑥 ∈ (Base‘𝑌))
35 simpr 476 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑤𝐼)
361, 27, 29, 31, 33, 34, 35prdsbasprj 16179 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑥𝑤) ∈ (Base‘(𝑅𝑤)))
37 simpr2 1088 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
3837adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑦 ∈ (Base‘𝑌))
391, 27, 29, 31, 33, 38, 35prdsbasprj 16179 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑦𝑤) ∈ (Base‘(𝑅𝑤)))
40 simpr3 1089 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
4140adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑧 ∈ (Base‘𝑌))
421, 27, 29, 31, 33, 41, 35prdsbasprj 16179 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))
43 eqid 2651 . . . . . . . . 9 (Base‘(𝑅𝑤)) = (Base‘(𝑅𝑤))
44 eqid 2651 . . . . . . . . 9 (+g‘(𝑅𝑤)) = (+g‘(𝑅𝑤))
45 eqid 2651 . . . . . . . . 9 (.r‘(𝑅𝑤)) = (.r‘(𝑅𝑤))
4643, 44, 45ringdi 18612 . . . . . . . 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))))
4726, 36, 39, 42, 46syl13anc 1368 . . . . . . 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))))
48 eqid 2651 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
491, 27, 29, 31, 33, 38, 41, 48, 35prdsplusgfval 16181 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(+g𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤)))
5049oveq2d 6706 . . . . . . 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))))
51 eqid 2651 . . . . . . . . 9 (.r𝑌) = (.r𝑌)
521, 27, 29, 31, 33, 34, 38, 51, 35prdsmulrfval 16183 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤)))
531, 27, 29, 31, 33, 34, 41, 51, 35prdsmulrfval 16183 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑧)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
5452, 53oveq12d 6708 . . . . . . 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))))
5547, 50, 543eqtr4d 2695 . . . . . 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)))
5655mpteq2dva 4777 . . . . 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))))
57 simpr1 1087 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
58 ringmnd 18602 . . . . . . . . . 10 (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
5958ssriv 3640 . . . . . . . . 9 Ring ⊆ Mnd
60 fss 6094 . . . . . . . . 9 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
614, 59, 60sylancl 695 . . . . . . . 8 (𝜑𝑅:𝐼⟶Mnd)
6261adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
631, 27, 48, 28, 30, 62, 37, 40prdsplusgcl 17368 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g𝑌)𝑧) ∈ (Base‘𝑌))
641, 27, 28, 30, 32, 57, 63, 51prdsmulrval 16182 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = (𝑤𝐼 ↦ ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤))))
651, 27, 51, 28, 30, 25, 57, 37prdsmulrcl 18657 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑦) ∈ (Base‘𝑌))
661, 27, 51, 28, 30, 25, 57, 40prdsmulrcl 18657 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑧) ∈ (Base‘𝑌))
671, 27, 28, 30, 32, 65, 66, 48prdsplusgval 16180 . . . . 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))))
6856, 64, 673eqtr4d 2695 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)))
6943, 44, 45ringdir 18613 . . . . . . . 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))))
7026, 36, 39, 42, 69syl13anc 1368 . . . . . . 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))))
711, 27, 29, 31, 33, 34, 38, 48, 35prdsplusgfval 16181 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(+g𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤)))
7271oveq1d 6705 . . . . . . 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)))
731, 27, 29, 31, 33, 38, 41, 51, 35prdsmulrfval 16183 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(.r𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
7453, 73oveq12d 6708 . . . . . . 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))))
7570, 72, 743eqtr4d 2695 . . . . . 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)))
7675mpteq2dva 4777 . . . . 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))))
771, 27, 48, 28, 30, 62, 57, 37prdsplusgcl 17368 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g𝑌)𝑦) ∈ (Base‘𝑌))
781, 27, 28, 30, 32, 77, 40, 51prdsmulrval 16182 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = (𝑤𝐼 ↦ (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
791, 27, 51, 28, 30, 25, 37, 40prdsmulrcl 18657 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r𝑌)𝑧) ∈ (Base‘𝑌))
801, 27, 28, 30, 32, 66, 79, 48prdsplusgval 16180 . . . . 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))))
8176, 78, 803eqtr4d 2695 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧)))
8268, 81jca 553 . . 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))))
8382ralrimivvva 3001 . 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))))
8427, 16, 48, 51isring 18597 . 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)))))
859, 24, 83, 84syl3anbrc 1265 1 (𝜑𝑌 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607  cmpt 4762  cres 5145  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Xscprds 16153  Mndcmnd 17341  Grpcgrp 17469  mulGrpcmgp 18535  Ringcrg 18593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-prds 16155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-mgp 18536  df-ring 18595
This theorem is referenced by:  prdscrngd  18659  pwsring  18661
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