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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 11211 | . . 3 โข (๐ โ 1 โ โ) | |
3 | adddirp1d.b | . . 3 โข (๐ โ ๐ต โ โ) | |
4 | 1, 2, 3 | adddird 11241 | . 2 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + (1 ยท ๐ต))) |
5 | 3 | mullidd 11234 | . . 3 โข (๐ โ (1 ยท ๐ต) = ๐ต) |
6 | 5 | oveq2d 7427 | . 2 โข (๐ โ ((๐ด ยท ๐ต) + (1 ยท ๐ต)) = ((๐ด ยท ๐ต) + ๐ต)) |
7 | 4, 6 | eqtrd 2772 | 1 โข (๐ โ ((๐ด + 1) ยท ๐ต) = ((๐ด ยท ๐ต) + ๐ต)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 (class class class)co 7411 โcc 11110 1c1 11113 + caddc 11115 ยท cmul 11117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-mulcl 11174 ax-mulcom 11176 ax-mulass 11178 ax-distr 11179 ax-1rid 11182 ax-cnre 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 |
This theorem is referenced by: modvalp1 13857 pcexp 16794 mulgnnass 18991 cnfldmulg 20983 dgrcolem1 25794 abelthlem2 25951 2lgsoddprmlem3d 26923 chpdifbndlem1 27063 breprexplemc 33713 fltnltalem 41486 lt3addmuld 44090 lt4addmuld 44095 itgsinexp 44750 fourierdlem19 44921 fourierdlem35 44937 fourierdlem51 44952 |
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