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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 20139 | . 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 494 | 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 394 โง w3a 1084 = wceq 1533 โ wcel 2098 โcfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 SRingcsrg 20133 |
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 2166 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-srg 20134 |
This theorem is referenced by: srgo2times 20159 srgcom4lem 20160 srgmulgass 20164 srgrmhm 20169 |
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