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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 7986 | . . 3 โข (๐ โ 1 โ โ) | |
3 | adddirp1d.b | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | 1, 2, 3 | adddird 7996 | . 2 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + (1 ยท ๐ต))) |
5 | 3 | mulid2d 7989 | . . 3 โข (๐ โ (1 ยท ๐ต) = ๐ต) |
6 | 5 | oveq2d 5904 | . 2 โข (๐ โ ((๐ด ยท ๐ต) + (1 ยท ๐ต)) = ((๐ด ยท ๐ต) + ๐ต)) |
7 | 4, 6 | eqtrd 2220 | 1 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + ๐ต)) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 = wceq 1363 โ wcel 2158 (class class class)co 5888 โcc 7822 1c1 7825 + caddc 7827 ยท cmul 7829 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-resscn 7916 ax-1cn 7917 ax-icn 7919 ax-addcl 7920 ax-mulcl 7922 ax-mulcom 7925 ax-mulass 7927 ax-distr 7928 ax-1rid 7931 ax-cnre 7935 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-iota 5190 df-fv 5236 df-ov 5891 |
This theorem is referenced by: modqvalp1 10356 hashxp 10819 fsumconst 11475 divalglemnqt 11938 pcexp 12322 mulgnnass 13049 cnfldmulg 13727 2lgsoddprmlem3d 14729 |
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