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Theorem prdsringd 18819
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 18759 . . . . 5 (𝑥 ∈ Ring → 𝑥 ∈ Grp)
65ssriv 3754 . . . 4 Ring ⊆ Grp
7 fss 6196 . . . 4 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 566 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17732 . 2 (𝜑𝑌 ∈ Grp)
10 eqid 2770 . . . 4 (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11 mgpf 18766 . . . . 5 (mulGrp ↾ Ring):Ring⟶Mnd
12 fco2 6199 . . . . 5 (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1311, 4, 12sylancr 567 . . . 4 (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1410, 2, 3, 13prdsmndd 17530 . . 3 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15 eqidd 2771 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16 eqid 2770 . . . . . 6 (mulGrp‘𝑌) = (mulGrp‘𝑌)
17 ffn 6185 . . . . . . 7 (𝑅:𝐼⟶Ring → 𝑅 Fn 𝐼)
184, 17syl 17 . . . . . 6 (𝜑𝑅 Fn 𝐼)
191, 16, 10, 2, 3, 18prdsmgp 18817 . . . . 5 (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
2019simpld 476 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
2119simprd 477 . . . . 5 (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
2221oveqdr 6818 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
2315, 20, 22mndpropd 17523 . . 3 (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
2414, 23mpbird 247 . 2 (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
254adantr 466 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
2625ffvelrnda 6502 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑅𝑤) ∈ Ring)
27 eqid 2770 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
283adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆𝑉)
2928adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑆𝑉)
302adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼𝑊)
3130adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝐼𝑊)
3218adantr 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
3332adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑅 Fn 𝐼)
34 simplr1 1259 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑥 ∈ (Base‘𝑌))
35 simpr 471 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑤𝐼)
361, 27, 29, 31, 33, 34, 35prdsbasprj 16339 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑥𝑤) ∈ (Base‘(𝑅𝑤)))
37 simpr2 1234 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
3837adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑦 ∈ (Base‘𝑌))
391, 27, 29, 31, 33, 38, 35prdsbasprj 16339 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑦𝑤) ∈ (Base‘(𝑅𝑤)))
40 simpr3 1236 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
4140adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑧 ∈ (Base‘𝑌))
421, 27, 29, 31, 33, 41, 35prdsbasprj 16339 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))
43 eqid 2770 . . . . . . . . 9 (Base‘(𝑅𝑤)) = (Base‘(𝑅𝑤))
44 eqid 2770 . . . . . . . . 9 (+g‘(𝑅𝑤)) = (+g‘(𝑅𝑤))
45 eqid 2770 . . . . . . . . 9 (.r‘(𝑅𝑤)) = (.r‘(𝑅𝑤))
4643, 44, 45ringdi 18773 . . . . . . . 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 1477 . . . . . . 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 2770 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
491, 27, 29, 31, 33, 38, 41, 48, 35prdsplusgfval 16341 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(+g𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤)))
5049oveq2d 6808 . . . . . . 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 2770 . . . . . . . . 9 (.r𝑌) = (.r𝑌)
521, 27, 29, 31, 33, 34, 38, 51, 35prdsmulrfval 16343 . . . . . . . 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 16343 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑧)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
5452, 53oveq12d 6810 . . . . . . 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 2814 . . . . . 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 4876 . . . . 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 1232 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
58 ringmnd 18763 . . . . . . . . . 10 (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
5958ssriv 3754 . . . . . . . . 9 Ring ⊆ Mnd
60 fss 6196 . . . . . . . . 9 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
614, 59, 60sylancl 566 . . . . . . . 8 (𝜑𝑅:𝐼⟶Mnd)
6261adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
631, 27, 48, 28, 30, 62, 37, 40prdsplusgcl 17528 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g𝑌)𝑧) ∈ (Base‘𝑌))
641, 27, 28, 30, 32, 57, 63, 51prdsmulrval 16342 . . . . 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 18818 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑦) ∈ (Base‘𝑌))
661, 27, 51, 28, 30, 25, 57, 40prdsmulrcl 18818 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑧) ∈ (Base‘𝑌))
671, 27, 28, 30, 32, 65, 66, 48prdsplusgval 16340 . . . . 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 2814 . . . 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 18774 . . . . . . . 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 1477 . . . . . . 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 16341 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(+g𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤)))
7271oveq1d 6807 . . . . . . 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 16343 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(.r𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
7453, 73oveq12d 6810 . . . . . . 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 2814 . . . . . 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 4876 . . . . 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 17528 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g𝑌)𝑦) ∈ (Base‘𝑌))
781, 27, 28, 30, 32, 77, 40, 51prdsmulrval 16342 . . . . 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 18818 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r𝑌)𝑧) ∈ (Base‘𝑌))
801, 27, 28, 30, 32, 66, 79, 48prdsplusgval 16340 . . . . 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 2814 . . . 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 495 . . 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 3120 . 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 18758 . 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 1427 1 (𝜑𝑌 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wss 3721  cmpt 4861  cres 5251  ccom 5253   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Xscprds 16313  Mndcmnd 17501  Grpcgrp 17629  mulGrpcmgp 18696  Ringcrg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-hom 16173  df-cco 16174  df-0g 16309  df-prds 16315  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-mgp 18697  df-ring 18756
This theorem is referenced by:  prdscrngd  18820  pwsring  18822
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