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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 11238 | . 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 1543 ∈ wcel 2112 (class class class)co 7191 ℂcc 10692 1c1 10695 · cmul 10699 -cneg 11028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-neg 11030 |
This theorem is referenced by: recextlem1 11427 ofnegsub 11793 modnegd 13464 modsumfzodifsn 13482 m1expcl2 13622 remullem 14656 sqrtneglem 14795 iseraltlem2 15211 iseraltlem3 15212 fsumneg 15314 incexclem 15363 incexc 15364 risefallfac 15549 efi4p 15661 cosadd 15689 absefib 15722 efieq1re 15723 pwp1fsum 15915 bitsinv1lem 15963 bezoutlem1 16062 pythagtriplem4 16335 negcncf 23773 mbfneg 24501 itg1sub 24561 itgcnlem 24641 i1fibl 24659 itgitg1 24660 itgmulc2 24685 dvmptneg 24817 dvlipcn 24845 lhop2 24866 logneg 25430 lognegb 25432 tanarg 25461 logtayl 25502 logtayl2 25504 asinlem 25705 asinlem2 25706 asinsin 25729 efiatan2 25754 2efiatan 25755 atandmtan 25757 atantan 25760 atans2 25768 dvatan 25772 basellem5 25921 lgsdir2lem4 26163 gausslemma2dlem5a 26205 lgseisenlem1 26210 lgseisenlem2 26211 rpvmasum2 26347 ostth3 26473 smcnlem 28732 ipval2 28742 dipsubdir 28883 his2sub 29127 qqhval2lem 31597 fwddifnp1 34153 itgmulc2nc 35531 ftc1anclem5 35540 areacirclem1 35551 lcmineqlem8 39727 negexpidd 40148 3cubeslem3r 40153 mzpsubmpt 40209 rmym1 40401 rngunsnply 40642 reabssgn 40861 sqrtcval 40866 expgrowth 41567 isumneg 42761 climneg 42769 stoweidlem22 43181 stirlinglem5 43237 fourierdlem97 43362 sqwvfourb 43388 etransclem46 43439 smfneg 43952 sharhght 43996 sigaradd 43997 altgsumbcALT 45305 |
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