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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 10509 | . 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 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 1c1 9975 · cmul 9979 -cneg 10305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-neg 10307 |
This theorem is referenced by: recextlem1 10695 ofnegsub 11056 modnegd 12765 modsumfzodifsn 12783 m1expcl2 12922 remullem 13912 sqrtneglem 14051 iseraltlem2 14457 iseraltlem3 14458 fsumneg 14563 incexclem 14612 incexc 14613 risefallfac 14799 efi4p 14911 cosadd 14939 absefib 14972 efieq1re 14973 pwp1fsum 15161 bitsinv1lem 15210 bezoutlem1 15303 pythagtriplem4 15571 negcncf 22768 mbfneg 23462 itg1sub 23521 itgcnlem 23601 i1fibl 23619 itgitg1 23620 itgmulc2 23645 dvmptneg 23774 dvlipcn 23802 lhop2 23823 logneg 24379 lognegb 24381 tanarg 24410 logtayl 24451 logtayl2 24453 asinlem 24640 asinlem2 24641 asinsin 24664 efiatan2 24689 2efiatan 24690 atandmtan 24692 atantan 24695 atans2 24703 dvatan 24707 basellem5 24856 lgsdir2lem4 25098 gausslemma2dlem5a 25140 lgseisenlem1 25145 lgseisenlem2 25146 rpvmasum2 25246 ostth3 25372 smcnlem 27680 ipval2 27690 dipsubdir 27831 his2sub 28077 qqhval2lem 30153 fwddifnp1 32397 itgmulc2nc 33608 ftc1anclem5 33619 areacirclem1 33630 mzpsubmpt 37623 rmym1 37817 rngunsnply 38060 expgrowth 38851 isumneg 40152 climneg 40160 stoweidlem22 40557 stirlinglem5 40613 fourierdlem97 40738 sqwvfourb 40764 etransclem46 40815 smfneg 41331 sharhght 41375 sigaradd 41376 altgsumbcALT 42456 |
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