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Mirrors > Home > ILE Home > Th. List > adddirp1d | GIF version |
Description: Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
adddirp1d.a | โข (๐ โ ๐ด โ โ) |
adddirp1d.b | โข (๐ โ ๐ต โ โ) |
Ref | Expression |
---|---|
adddirp1d | โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + ๐ต)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | adddirp1d.a | . . 3 โข (๐ โ ๐ด โ โ) | |
2 | 1cnd 7992 | . . 3 โข (๐ โ 1 โ โ) | |
3 | adddirp1d.b | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | 1, 2, 3 | adddird 8002 | . 2 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + (1 ยท ๐ต))) |
5 | 3 | mulid2d 7995 | . . 3 โข (๐ โ (1 ยท ๐ต) = ๐ต) |
6 | 5 | oveq2d 5907 | . 2 โข (๐ โ ((๐ด ยท ๐ต) + (1 ยท ๐ต)) = ((๐ด ยท ๐ต) + ๐ต)) |
7 | 4, 6 | eqtrd 2222 | 1 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + ๐ต)) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 = wceq 1364 โ wcel 2160 (class class class)co 5891 โcc 7828 1c1 7831 + caddc 7833 ยท cmul 7835 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-resscn 7922 ax-1cn 7923 ax-icn 7925 ax-addcl 7926 ax-mulcl 7928 ax-mulcom 7931 ax-mulass 7933 ax-distr 7934 ax-1rid 7937 ax-cnre 7941 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5193 df-fv 5239 df-ov 5894 |
This theorem is referenced by: modqvalp1 10362 hashxp 10825 fsumconst 11481 divalglemnqt 11944 pcexp 12328 mulgnnass 13069 cnfldmulg 13846 2lgsoddprmlem3d 14861 |
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