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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 | โข ๐ต = (Baseโ๐ ) |
ringdi.p | โข + = (+gโ๐ ) |
ringdi.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
ringdi | 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))) |
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 20072 | . 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 | simpld 496 | 1 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))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 โง w3a 1088 = wceq 1542 โ wcel 2107 โcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 .rcmulr 17198 Ringcrg 20056 |
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 5307 |
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 2942 df-ral 3063 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 df-ring 20058 |
This theorem is referenced by: ringcomlem 20096 ringrz 20108 ringnegr 20115 ringsubdi 20119 ringlghm 20124 prdsringd 20134 imasring 20143 opprring 20161 issubrg2 20339 cntzsubr 20353 sralmod 20809 psrlmod 21521 psrdi 21526 mamudir 21904 mdetrlin 22104 mdetuni0 22123 ply1divex 25654 lfladdcl 37941 lflvsdi2 37949 dvhlveclem 39979 |
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