MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdsrngd Structured version   Visualization version   GIF version

Theorem prdsrngd 20174
Description: A product of non-unital rings is a non-unital ring. (Contributed by AV, 22-Feb-2025.)
Hypotheses
Ref Expression
prdsrngd.y 𝑌 = (𝑆Xs𝑅)
prdsrngd.i (𝜑𝐼𝑊)
prdsrngd.s (𝜑𝑆𝑉)
prdsrngd.r (𝜑𝑅:𝐼⟶Rng)
Assertion
Ref Expression
prdsrngd (𝜑𝑌 ∈ Rng)

Proof of Theorem prdsrngd
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsrngd.y . . 3 𝑌 = (𝑆Xs𝑅)
2 prdsrngd.i . . 3 (𝜑𝐼𝑊)
3 prdsrngd.s . . 3 (𝜑𝑆𝑉)
4 prdsrngd.r . . . 4 (𝜑𝑅:𝐼⟶Rng)
5 rngabl 20153 . . . . 5 (𝑥 ∈ Rng → 𝑥 ∈ Abel)
65ssriv 3986 . . . 4 Rng ⊆ Abel
7 fss 6751 . . . 4 ((𝑅:𝐼⟶Rng ∧ Rng ⊆ Abel) → 𝑅:𝐼⟶Abel)
84, 6, 7sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Abel)
91, 2, 3, 8prdsabld 19881 . 2 (𝜑𝑌 ∈ Abel)
10 eqid 2736 . . . 4 (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅))
11 rngmgpf 20155 . . . . 5 (mulGrp ↾ Rng):Rng⟶Smgrp
12 fco2 6761 . . . . 5 (((mulGrp ↾ Rng):Rng⟶Smgrp ∧ 𝑅:𝐼⟶Rng) → (mulGrp ∘ 𝑅):𝐼⟶Smgrp)
1311, 4, 12sylancr 587 . . . 4 (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Smgrp)
1410, 2, 3, 13prdssgrpd 18747 . . 3 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Smgrp)
15 fvexd 6920 . . . 4 (𝜑 → (mulGrp‘𝑌) ∈ V)
16 ovexd 7467 . . . 4 (𝜑 → (𝑆Xs(mulGrp ∘ 𝑅)) ∈ V)
17 eqidd 2737 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌)))
18 eqid 2736 . . . . . 6 (mulGrp‘𝑌) = (mulGrp‘𝑌)
194ffnd 6736 . . . . . 6 (𝜑𝑅 Fn 𝐼)
201, 18, 10, 2, 3, 19prdsmgp 20149 . . . . 5 (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))))
2120simpld 494 . . . 4 (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))))
2220simprd 495 . . . . 5 (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))
2322oveqdr 7460 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦))
2415, 16, 17, 21, 23sgrppropd 18745 . . 3 (𝜑 → ((mulGrp‘𝑌) ∈ Smgrp ↔ (𝑆Xs(mulGrp ∘ 𝑅)) ∈ Smgrp))
2514, 24mpbird 257 . 2 (𝜑 → (mulGrp‘𝑌) ∈ Smgrp)
264adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Rng)
2726ffvelcdmda 7103 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑅𝑤) ∈ Rng)
28 eqid 2736 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
293adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑆𝑉)
3029adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑆𝑉)
312adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝐼𝑊)
3231adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝐼𝑊)
3319adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅 Fn 𝐼)
3433adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑅 Fn 𝐼)
35 simplr1 1215 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑥 ∈ (Base‘𝑌))
36 simpr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑤𝐼)
371, 28, 30, 32, 34, 35, 36prdsbasprj 17518 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑥𝑤) ∈ (Base‘(𝑅𝑤)))
38 simpr2 1195 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑦 ∈ (Base‘𝑌))
3938adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑦 ∈ (Base‘𝑌))
401, 28, 30, 32, 34, 39, 36prdsbasprj 17518 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑦𝑤) ∈ (Base‘(𝑅𝑤)))
41 simpr3 1196 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑧 ∈ (Base‘𝑌))
4241adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → 𝑧 ∈ (Base‘𝑌))
431, 28, 30, 32, 34, 42, 36prdsbasprj 17518 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → (𝑧𝑤) ∈ (Base‘(𝑅𝑤)))
44 eqid 2736 . . . . . . . . 9 (Base‘(𝑅𝑤)) = (Base‘(𝑅𝑤))
45 eqid 2736 . . . . . . . . 9 (+g‘(𝑅𝑤)) = (+g‘(𝑅𝑤))
46 eqid 2736 . . . . . . . . 9 (.r‘(𝑅𝑤)) = (.r‘(𝑅𝑤))
4744, 45, 46rngdi 20158 . . . . . . . 8 (((𝑅𝑤) ∈ Rng ∧ ((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))))
4827, 37, 40, 43, 47syl13anc 1373 . . . . . . 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))))
49 eqid 2736 . . . . . . . . 9 (+g𝑌) = (+g𝑌)
501, 28, 30, 32, 34, 39, 42, 49, 36prdsplusgfval 17520 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(+g𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(+g‘(𝑅𝑤))(𝑧𝑤)))
5150oveq2d 7448 . . . . . . 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))))
52 eqid 2736 . . . . . . . . 9 (.r𝑌) = (.r𝑌)
531, 28, 30, 32, 34, 35, 39, 52, 36prdsmulrfval 17522 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑦𝑤)))
541, 28, 30, 32, 34, 35, 42, 52, 36prdsmulrfval 17522 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(.r𝑌)𝑧)‘𝑤) = ((𝑥𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
5553, 54oveq12d 7450 . . . . . . 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))))
5648, 51, 553eqtr4d 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)))
5756mpteq2dva 5241 . . . . 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))))
58 simpr1 1194 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑥 ∈ (Base‘𝑌))
59 rnggrp 20156 . . . . . . . . . . 11 (𝑥 ∈ Rng → 𝑥 ∈ Grp)
6059grpmndd 18965 . . . . . . . . . 10 (𝑥 ∈ Rng → 𝑥 ∈ Mnd)
6160ssriv 3986 . . . . . . . . 9 Rng ⊆ Mnd
62 fss 6751 . . . . . . . . 9 ((𝑅:𝐼⟶Rng ∧ Rng ⊆ Mnd) → 𝑅:𝐼⟶Mnd)
634, 61, 62sylancl 586 . . . . . . . 8 (𝜑𝑅:𝐼⟶Mnd)
6463adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → 𝑅:𝐼⟶Mnd)
651, 28, 49, 29, 31, 64, 38, 41prdsplusgcl 18782 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(+g𝑌)𝑧) ∈ (Base‘𝑌))
661, 28, 29, 31, 33, 58, 65, 52prdsmulrval 17521 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = (𝑤𝐼 ↦ ((𝑥𝑤)(.r‘(𝑅𝑤))((𝑦(+g𝑌)𝑧)‘𝑤))))
671, 28, 52, 29, 31, 26, 58, 38prdsmulrngcl 20173 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑦) ∈ (Base‘𝑌))
681, 28, 52, 29, 31, 26, 58, 41prdsmulrngcl 20173 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)𝑧) ∈ (Base‘𝑌))
691, 28, 29, 31, 33, 67, 68, 49prdsplusgval 17519 . . . . 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))))
7057, 66, 693eqtr4d 2786 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(.r𝑌)(𝑦(+g𝑌)𝑧)) = ((𝑥(.r𝑌)𝑦)(+g𝑌)(𝑥(.r𝑌)𝑧)))
7144, 45, 46rngdir 20159 . . . . . . . 8 (((𝑅𝑤) ∈ Rng ∧ ((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))))
7227, 37, 40, 43, 71syl13anc 1373 . . . . . . 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))))
731, 28, 30, 32, 34, 35, 39, 49, 36prdsplusgfval 17520 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑥(+g𝑌)𝑦)‘𝑤) = ((𝑥𝑤)(+g‘(𝑅𝑤))(𝑦𝑤)))
7473oveq1d 7447 . . . . . . 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)))
751, 28, 30, 32, 34, 39, 42, 52, 36prdsmulrfval 17522 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) ∧ 𝑤𝐼) → ((𝑦(.r𝑌)𝑧)‘𝑤) = ((𝑦𝑤)(.r‘(𝑅𝑤))(𝑧𝑤)))
7654, 75oveq12d 7450 . . . . . . 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))))
7772, 74, 763eqtr4d 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)))
7877mpteq2dva 5241 . . . . 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))))
791, 28, 49, 29, 31, 64, 58, 38prdsplusgcl 18782 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑥(+g𝑌)𝑦) ∈ (Base‘𝑌))
801, 28, 29, 31, 33, 79, 41, 52prdsmulrval 17521 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = (𝑤𝐼 ↦ (((𝑥(+g𝑌)𝑦)‘𝑤)(.r‘(𝑅𝑤))(𝑧𝑤))))
811, 28, 52, 29, 31, 26, 38, 41prdsmulrngcl 20173 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → (𝑦(.r𝑌)𝑧) ∈ (Base‘𝑌))
821, 28, 29, 31, 33, 68, 81, 49prdsplusgval 17519 . . . . 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))))
8378, 80, 823eqtr4d 2786 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝑌) ∧ 𝑦 ∈ (Base‘𝑌) ∧ 𝑧 ∈ (Base‘𝑌))) → ((𝑥(+g𝑌)𝑦)(.r𝑌)𝑧) = ((𝑥(.r𝑌)𝑧)(+g𝑌)(𝑦(.r𝑌)𝑧)))
8470, 83jca 511 . . 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))))
8584ralrimivvva 3204 . 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))))
8628, 18, 49, 52isrng 20152 . 2 (𝑌 ∈ Rng ↔ (𝑌 ∈ Abel ∧ (mulGrp‘𝑌) ∈ Smgrp ∧ ∀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)))))
879, 25, 85, 86syl3anbrc 1343 1 (𝜑𝑌 ∈ Rng)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  Vcvv 3479  wss 3950  cmpt 5224  cres 5686  ccom 5688   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  Xscprds 17491  Smgrpcsgrp 18732  Mndcmnd 18748  Abelcabl 19800  mulGrpcmgp 20138  Rngcrng 20150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-prds 17493  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151
This theorem is referenced by:  xpsrngd  20177
  Copyright terms: Public domain W3C validator