![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 11168 | . . 3 โข 1 โ โ | |
2 | mulneg1 11650 | . . 3 โข ((1 โ โ โง ๐ด โ โ) โ (-1 ยท ๐ด) = -(1 ยท ๐ด)) | |
3 | 1, 2 | mpan 689 | . 2 โข (๐ด โ โ โ (-1 ยท ๐ด) = -(1 ยท ๐ด)) |
4 | mullid 11213 | . . 3 โข (๐ด โ โ โ (1 ยท ๐ด) = ๐ด) | |
5 | 4 | negeqd 11454 | . 2 โข (๐ด โ โ โ -(1 ยท ๐ด) = -๐ด) |
6 | 3, 5 | eqtrd 2773 | 1 โข (๐ด โ โ โ (-1 ยท ๐ด) = -๐ด) |
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 ยท cmul 11115 -cneg 11445 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 |
This theorem is referenced by: addneg1mul 11656 mulm1i 11659 mulm1d 11666 div2neg 11937 sqrtneglem 15213 sqreulem 15306 sinhval 16097 coshval 16098 demoivreALT 16144 sinmpi 25997 cosmpi 25998 sinppi 25999 cosppi 26000 cxpsqrt 26211 relogbdiv 26284 angneg 26308 lgsdir2lem4 26831 cnnvm 29935 cncph 30072 hvm1neg 30285 hvsubdistr2 30303 lnfnsubi 31299 dvasin 36572 lcmineqlem1 40894 |
Copyright terms: Public domain | W3C validator |