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Mirrors > Home > MPE Home > Th. List > srgdi | 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 |
---|---|
srgdi | 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))) |
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 20131 | . 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 | simpld 494 | 1 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))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 โง w3a 1085 = wceq 1534 โ wcel 2099 โcfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 .rcmulr 17233 SRingcsrg 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-srg 20126 |
This theorem is referenced by: srgcom4lem 20152 srglmhm 20160 srgbinomlem 20169 |
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