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Mirrors > Home > MPE Home > Th. List > ringdi | Structured version Visualization version GIF version |
Description: Distributive law for the multiplication operation of a ring (left-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 |
---|---|
ringdi | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 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 | ringi 19306 | . 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 | simpld 498 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Ringcrg 19290 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-ring 19292 |
This theorem is referenced by: ringcom 19325 ringrz 19334 rngnegr 19341 ringsubdi 19345 ringlghm 19350 prdsringd 19358 imasring 19365 opprring 19377 issubrg2 19548 cntzsubr 19561 sralmod 19952 psrlmod 20639 psrdi 20644 mamudir 21009 mdetrlin 21207 mdetuni0 21226 ply1divex 24737 lfladdcl 36367 lflvsdi2 36375 dvhlveclem 38404 lidlrng 44551 |
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