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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 19310 | . 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 498 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Ringcrg 19297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-ring 19299 |
This theorem is referenced by: ringadd2 19325 rngo2times 19326 ringcom 19329 ringlz 19337 ringnegl 19344 rngsubdir 19350 mulgass2 19351 ringrghm 19355 prdsringd 19362 imasring 19369 opprring 19381 issubrg2 19555 cntzsubr 19568 sralmod 19959 psrlmod 20181 psrdir 20187 evlslem1 20295 frlmphl 20925 mamudi 21012 mdetrlin 21211 dvrdir 30861 mxidlprm 30977 lflvscl 36228 lflvsdi1 36229 dvhlveclem 38259 lidlrng 44218 |
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