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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 | ringi 19435 | . 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 499 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 +gcplusg 16671 .rcmulr 16672 Ringcrg 19419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-iota 6298 df-fv 6348 df-ov 7176 df-ring 19421 |
This theorem is referenced by: ringadd2 19450 rngo2times 19451 ringcom 19454 ringlz 19462 ringnegl 19469 rngsubdir 19475 mulgass2 19476 ringrghm 19480 prdsringd 19487 imasring 19494 opprring 19506 issubrg2 19677 cntzsubr 19690 sralmod 20081 frlmphl 20600 psrlmod 20783 psrdir 20789 evlslem1 20899 mamudi 21157 mdetrlin 21356 dvrdir 31067 mxidlprm 31215 lflvscl 36737 lflvsdi1 36738 dvhlveclem 38768 lidlrng 45049 |
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