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Mirrors > Home > MPE Home > Th. List > adddirp1d | Structured version Visualization version 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 11209 | . . 3 โข (๐ โ 1 โ โ) | |
3 | adddirp1d.b | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | 1, 2, 3 | adddird 11239 | . 2 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + (1 ยท ๐ต))) |
5 | 3 | mullidd 11232 | . . 3 โข (๐ โ (1 ยท ๐ต) = ๐ต) |
6 | 5 | oveq2d 7425 | . 2 โข (๐ โ ((๐ด ยท ๐ต) + (1 ยท ๐ต)) = ((๐ด ยท ๐ต) + ๐ต)) |
7 | 4, 6 | eqtrd 2773 | 1 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 (class class class)co 7409 โcc 11108 1c1 11111 + caddc 11113 ยท cmul 11115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-mulcl 11172 ax-mulcom 11174 ax-mulass 11176 ax-distr 11177 ax-1rid 11180 ax-cnre 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-iota 6496 df-fv 6552 df-ov 7412 |
This theorem is referenced by: modvalp1 13855 pcexp 16792 mulgnnass 18989 cnfldmulg 20977 dgrcolem1 25787 abelthlem2 25944 2lgsoddprmlem3d 26916 chpdifbndlem1 27056 breprexplemc 33644 fltnltalem 41404 lt3addmuld 44011 lt4addmuld 44016 itgsinexp 44671 fourierdlem19 44842 fourierdlem35 44858 fourierdlem51 44873 |
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