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Mirrors > Home > MPE Home > Th. List > mulm1 | Structured version Visualization version GIF version |
Description: Product with minus one is negative. (Contributed by NM, 16-Nov-1999.) |
Ref | Expression |
---|---|
mulm1 | โข (๐ด โ โ โ (-1 ยท ๐ด) = -๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11164 | . . 3 โข 1 โ โ | |
2 | mulneg1 11646 | . . 3 โข ((1 โ โ โง ๐ด โ โ) โ (-1 ยท ๐ด) = -(1 ยท ๐ด)) | |
3 | 1, 2 | mpan 688 | . 2 โข (๐ด โ โ โ (-1 ยท ๐ด) = -(1 ยท ๐ด)) |
4 | mullid 11209 | . . 3 โข (๐ด โ โ โ (1 ยท ๐ด) = ๐ด) | |
5 | 4 | negeqd 11450 | . 2 โข (๐ด โ โ โ -(1 ยท ๐ด) = -๐ด) |
6 | 3, 5 | eqtrd 2772 | 1 โข (๐ด โ โ โ (-1 ยท ๐ด) = -๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 (class class class)co 7405 โcc 11104 1c1 11107 ยท cmul 11111 -cneg 11441 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 |
This theorem is referenced by: addneg1mul 11652 mulm1i 11655 mulm1d 11662 div2neg 11933 sqrtneglem 15209 sqreulem 15302 sinhval 16093 coshval 16094 demoivreALT 16140 sinmpi 25988 cosmpi 25989 sinppi 25990 cosppi 25991 cxpsqrt 26202 relogbdiv 26273 angneg 26297 lgsdir2lem4 26820 cnnvm 29922 cncph 30059 hvm1neg 30272 hvsubdistr2 30290 lnfnsubi 31286 dvasin 36560 lcmineqlem1 40882 |
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