![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20267 | . 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 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Ringcrg 20251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-ring 20253 |
This theorem is referenced by: ringo2times 20289 ringcomlem 20293 ringnegl 20316 mulgass2 20323 ringrghm 20327 prdsringd 20335 imasring 20344 dvrdir 20429 issubrg2 20609 cntzsubr 20623 sralmod 21212 frlmphl 21819 psrlmod 21998 psrdir 22004 evlslem1 22124 mamudi 22423 mdetrlin 22624 ringdird 33220 rlocaddval 33255 mxidlprm 33478 q1pdir 33603 r1pcyc 33607 lflvscl 39059 lflvsdi1 39060 dvhlveclem 41091 |
Copyright terms: Public domain | W3C validator |