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Mirrors > Home > MPE Home > Th. List > srgdir | Structured version Visualization version GIF version |
Description: Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgdi.b | โข ๐ต = (Baseโ๐ ) |
srgdi.p | โข + = (+gโ๐ ) |
srgdi.t | โข ยท = (.rโ๐ ) |
Ref | Expression |
---|---|
srgdir | โข ((๐ โ SRing 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 | srgdi.b | . . 3 โข ๐ต = (Baseโ๐ ) | |
2 | srgdi.p | . . 3 โข + = (+gโ๐ ) | |
3 | srgdi.t | . . 3 โข ยท = (.rโ๐ ) | |
4 | 1, 2, 3 | srgdilem 20095 | . 2 โข ((๐ โ SRing 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 495 | 1 โข ((๐ โ SRing 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 395 โง w3a 1084 = wceq 1533 โ wcel 2098 โcfv 6536 (class class class)co 7404 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 SRingcsrg 20089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-srg 20090 |
This theorem is referenced by: srgo2times 20115 srgcom4lem 20116 srgmulgass 20120 srgrmhm 20125 |
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