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Theorem prdsringd 20036
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 19969 . . . . 5 (𝑥 ∈ Ring → 𝑥 ∈ Grp)
65ssriv 3948 . . . 4 Ring ⊆ Grp
7 fss 6685 . . . 4 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 18857 . 2 (𝜑𝑌 ∈ Grp)
10 eqid 2736 . . . 4 (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11 mgpf 19979 . . . . 5 (mulGrp ↾ Ring):Ring⟶Mnd
12 fco2 6695 . . . . 5 (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1311, 4, 12sylancr 587 . . . 4 (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd)
1410, 2, 3, 13prdsmndd 18589 . . 3 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd)
15 eqidd 2737 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
16 eqid 2736 . . . . . 6 (mulGrp‘𝑌) = (mulGrp‘𝑌)
174ffnd 6669 . . . . . 6 (𝜑𝑅 Fn 𝐼)
181, 16, 10, 2, 3, 17prdsmgp 20034 . . . . 5 (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
1918simpld 495 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
2018simprd 496 . . . . 5 (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
2120oveqdr 7385 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
2215, 19, 21mndpropd 18581 . . 3 (𝜑 → ((mulGrp‘𝑌) ∈ Mnd ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Mnd))
2314, 22mpbird 256 . 2 (𝜑 → (mulGrp‘𝑌) ∈ Mnd)
244adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Ring)
2524ffvelcdmda 7035 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑅𝑤) ∈ Ring)
26 eqid 2736 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
273adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆𝑉)
2827adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑆𝑉)
292adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼𝑊)
3029adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝐼𝑊)
3117adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
3231adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑅 Fn 𝐼)
33 simplr1 1215 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑥 ∈ (Base‘𝑌))
34 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑤𝐼)
351, 26, 28, 30, 32, 33, 34prdsbasprj 17354 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑥𝑤) ∈ (Base‘(𝑅𝑤)))
36 simpr2 1195 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
3736adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑦 ∈ (Base‘𝑌))
381, 26, 28, 30, 32, 37, 34prdsbasprj 17354 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑦𝑤) ∈ (Base‘(𝑅𝑤)))
39 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
4039adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑧 ∈ (Base‘𝑌))
411, 26, 28, 30, 32, 40, 34prdsbasprj 17354 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))
42 eqid 2736 . . . . . . . . 9 (Base‘(𝑅𝑤)) = (Base‘(𝑅𝑤))
43 eqid 2736 . . . . . . . . 9 (+g‘(𝑅𝑤)) = (+g‘(𝑅𝑤))
44 eqid 2736 . . . . . . . . 9 (.r‘(𝑅𝑤)) = (.r‘(𝑅𝑤))
4542, 43, 44ringdi 19987 . . . . . . . 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 1372 . . . . . . 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 2736 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
481, 26, 28, 30, 32, 37, 40, 47, 34prdsplusgfval 17356 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(+g𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤)))
4948oveq2d 7373 . . . . . . 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 2736 . . . . . . . . 9 (.r𝑌) = (.r𝑌)
511, 26, 28, 30, 32, 33, 37, 50, 34prdsmulrfval 17358 . . . . . . . 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 17358 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑧)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
5351, 52oveq12d 7375 . . . . . . 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 2786 . . . . . 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 5205 . . . . 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 1194 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
57 ringmnd 19974 . . . . . . . . . 10 (𝑥 ∈ Ring → 𝑥 ∈ Mnd)
5857ssriv 3948 . . . . . . . . 9 Ring ⊆ Mnd
59 fss 6685 . . . . . . . . 9 ((𝑅:𝐼⟶Ring ∧ Ring ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
604, 58, 59sylancl 586 . . . . . . . 8 (𝜑𝑅:𝐼⟶Mnd)
6160adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
621, 26, 47, 27, 29, 61, 36, 39prdsplusgcl 18587 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g𝑌)𝑧) ∈ (Base‘𝑌))
631, 26, 27, 29, 31, 56, 62, 50prdsmulrval 17357 . . . . 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 20035 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑦) ∈ (Base‘𝑌))
651, 26, 50, 27, 29, 24, 56, 39prdsmulrcl 20035 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑧) ∈ (Base‘𝑌))
661, 26, 27, 29, 31, 64, 65, 47prdsplusgval 17355 . . . . 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 2786 . . . 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 19988 . . . . . . . 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 1372 . . . . . . 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 17356 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(+g𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤)))
7170oveq1d 7372 . . . . . . 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 17358 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(.r𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
7352, 72oveq12d 7375 . . . . . . 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 2786 . . . . . 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 5205 . . . . 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 18587 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g𝑌)𝑦) ∈ (Base‘𝑌))
771, 26, 27, 29, 31, 76, 39, 50prdsmulrval 17357 . . . . 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 20035 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r𝑌)𝑧) ∈ (Base‘𝑌))
791, 26, 27, 29, 31, 65, 78, 47prdsplusgval 17355 . . . . 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 2786 . . . 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 512 . . 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 3200 . 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 19968 . 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 1343 1 (𝜑𝑌 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910  cmpt 5188  cres 5635  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  Xscprds 17327  Mndcmnd 18556  Grpcgrp 18748  mulGrpcmgp 19896  Ringcrg 19964
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-prds 17329  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mgp 19897  df-ring 19966
This theorem is referenced by:  prdscrngd  20037  pwsring  20039
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