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Mirrors > Home > MPE Home > Th. List > muls1d | Structured version Visualization version GIF version |
Description: Multiplication by one minus a number. (Contributed by Scott Fenton, 23-Dec-2017.) |
Ref | Expression |
---|---|
muls1d.1 | โข (๐ โ ๐ด โ โ) |
muls1d.2 | โข (๐ โ ๐ต โ โ) |
Ref | Expression |
---|---|
muls1d | โข (๐ โ (๐ด ยท (๐ต โ 1)) = ((๐ด ยท ๐ต) โ ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | muls1d.1 | . . 3 โข (๐ โ ๐ด โ โ) | |
2 | muls1d.2 | . . 3 โข (๐ โ ๐ต โ โ) | |
3 | 1cnd 11240 | . . 3 โข (๐ โ 1 โ โ) | |
4 | 1, 2, 3 | subdid 11701 | . 2 โข (๐ โ (๐ด ยท (๐ต โ 1)) = ((๐ด ยท ๐ต) โ (๐ด ยท 1))) |
5 | 1 | mulridd 11262 | . . 3 โข (๐ โ (๐ด ยท 1) = ๐ด) |
6 | 5 | oveq2d 7436 | . 2 โข (๐ โ ((๐ด ยท ๐ต) โ (๐ด ยท 1)) = ((๐ด ยท ๐ต) โ ๐ด)) |
7 | 4, 6 | eqtrd 2768 | 1 โข (๐ โ (๐ด ยท (๐ต โ 1)) = ((๐ด ยท ๐ต) โ ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 (class class class)co 7420 โcc 11137 1c1 11140 ยท cmul 11144 โ cmin 11475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 |
This theorem is referenced by: 3dvds 16308 wilthlem2 27014 bposlem1 27230 lcmineqlem23 41522 fmtnorec2lem 46882 |
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