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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 | โข ๐ต = (Baseโ๐ ) |
ringdi.p | โข + = (+gโ๐ ) |
ringdi.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
ringdir | โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ + ๐) ยท ๐) = ((๐ ยท ๐) + (๐ ยท ๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringdi.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | ringdi.p | . . 3 โข + = (+gโ๐ ) | |
3 | ringdi.t | . . 3 โข ยท = (.rโ๐ ) | |
4 | 1, 2, 3 | ringdilem 19988 | . 2 โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ ยท (๐ + ๐)) = ((๐ ยท ๐) + (๐ ยท ๐)) โง ((๐ + ๐) ยท ๐) = ((๐ ยท ๐) + (๐ ยท ๐)))) |
5 | 4 | simprd 497 | 1 โข ((๐ โ Ring โง (๐ โ ๐ต โง ๐ โ ๐ต โง ๐ โ ๐ต)) โ ((๐ + ๐) ยท ๐) = ((๐ ยท ๐) + (๐ ยท ๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 โง w3a 1088 = wceq 1542 โ wcel 2107 โcfv 6500 (class class class)co 7361 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 Ringcrg 19972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 df-ring 19974 |
This theorem is referenced by: ringo2times 20004 ringcomlem 20008 ringlz 20019 ringnegl 20026 ringsubdir 20032 mulgass2 20033 ringrghm 20037 prdsringd 20044 imasring 20053 opprring 20068 dvrdir 20131 issubrg2 20284 cntzsubr 20298 sralmod 20701 frlmphl 21210 psrlmod 21393 psrdir 21399 evlslem1 21515 mamudi 21773 mdetrlin 21974 mxidlprm 32292 lflvscl 37589 lflvsdi1 37590 dvhlveclem 39621 lidlrng 46315 |
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