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| Mirrors > Home > MPE Home > Th. List > ringdir | Structured version Visualization version GIF version | ||
| Description: Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) |
| Ref | Expression |
|---|---|
| ringdi.b | ⊢ 𝐵 = (Base‘𝑅) |
| ringdi.p | ⊢ + = (+g‘𝑅) |
| ringdi.t | ⊢ · = (.r‘𝑅) |
| Ref | Expression |
|---|---|
| ringdir | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ringdi.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | ringdi.p | . . 3 ⊢ + = (+g‘𝑅) | |
| 3 | ringdi.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 4 | 1, 2, 3 | ringdilem 20246 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))) |
| 5 | 4 | simprd 495 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Ringcrg 20230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-ring 20232 |
| This theorem is referenced by: ringdird 20261 ringo2times 20272 ringcomlem 20276 ringnegl 20299 mulgass2 20306 ringrghm 20310 prdsringd 20318 imasring 20327 dvrdir 20412 issubrg2 20592 cntzsubr 20606 sralmod 21194 frlmphl 21801 psrlmod 21980 psrdir 21986 evlslem1 22106 mamudi 22407 mdetrlin 22608 rlocaddval 33272 mxidlprm 33498 q1pdir 33623 r1pcyc 33627 lflvscl 39078 lflvsdi1 39079 dvhlveclem 41110 |
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