| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > mulm1d | Structured version Visualization version GIF version | ||
| Description: Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| mulm1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| Ref | Expression |
|---|---|
| mulm1d | ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulm1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | mulm1 11704 | . 2 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (-1 · 𝐴) = -𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 1c1 11156 · cmul 11160 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: recextlem1 11893 ofnegsub 12264 modnegd 13967 modsumfzodifsn 13985 m1expcl2 14126 remullem 15167 sqrtneglem 15305 iseraltlem2 15719 iseraltlem3 15720 fsumneg 15823 incexclem 15872 incexc 15873 risefallfac 16060 efi4p 16173 cosadd 16201 absefib 16234 efieq1re 16235 pwp1fsum 16428 bitsinv1lem 16478 bezoutlem1 16576 pythagtriplem4 16857 negcncf 24948 negcncfOLD 24949 mbfneg 25685 itg1sub 25744 itgcnlem 25825 i1fibl 25843 itgitg1 25844 itgmulc2 25869 dvmptneg 26004 dvlipcn 26033 lhop2 26054 logneg 26630 lognegb 26632 tanarg 26661 logtayl 26702 logtayl2 26704 asinlem 26911 asinlem2 26912 asinsin 26935 efiatan2 26960 2efiatan 26961 atandmtan 26963 atantan 26966 atans2 26974 dvatan 26978 basellem5 27128 lgsdir2lem4 27372 gausslemma2dlem5a 27414 lgseisenlem1 27419 lgseisenlem2 27420 rpvmasum2 27556 ostth3 27682 smcnlem 30716 ipval2 30726 dipsubdir 30867 his2sub 31111 quad3d 32754 qqhval2lem 33982 fwddifnp1 36166 itgmulc2nc 37695 ftc1anclem5 37704 areacirclem1 37715 lcmineqlem8 42037 readvrec 42392 negexpidd 42693 3cubeslem3r 42698 mzpsubmpt 42754 rmym1 42947 rngunsnply 43181 reabssgn 43649 sqrtcval 43654 expgrowth 44354 isumneg 45617 climneg 45625 stoweidlem22 46037 stirlinglem5 46093 fourierdlem97 46218 sqwvfourb 46244 etransclem46 46295 smfneg 46818 sharhght 46880 sigaradd 46881 altgsumbcALT 48269 |
| Copyright terms: Public domain | W3C validator |