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Mirrors > Home > ILE Home > Th. List > srgdi | 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 13157 | . 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 112 | 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 set class |
Syntax hints: โ wi 4 โง wa 104 โง w3a 978 = wceq 1353 โ wcel 2148 โcfv 5218 (class class class)co 5877 Basecbs 12464 +gcplusg 12538 .rcmulr 12539 SRingcsrg 13151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-riota 5833 df-ov 5880 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-base 12470 df-plusg 12551 df-mulr 12552 df-0g 12712 df-srg 13152 |
This theorem is referenced by: srglmhm 13181 |
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