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Mirrors > Home > MPE Home > Th. List > neg1rr | Structured version Visualization version GIF version |
Description: -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.) |
Ref | Expression |
---|---|
neg1rr | ⊢ -1 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10363 | . 2 ⊢ 1 ∈ ℝ | |
2 | 1 | renegcli 10670 | 1 ⊢ -1 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 ℝcr 10258 1c1 10260 -cneg 10593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 df-neg 10595 |
This theorem is referenced by: dfceil2 12942 bernneq 13291 crre 14238 remim 14241 iseraltlem2 14797 iseraltlem3 14798 iseralt 14799 tanhbnd 15270 sinbnd2 15291 cosbnd2 15292 psgnodpmr 20302 xrhmeo 23122 xrhmph 23123 vitalilem2 23782 vitalilem4 23784 vitali 23786 mbfneg 23823 i1fsub 23881 itg1sub 23882 i1fibl 23980 itgitg1 23981 recosf1o 24688 efif1olem3 24697 relogbdiv 24926 ang180lem3 24958 1cubrlem 24988 atanre 25032 acosrecl 25050 atandmcj 25056 leibpilem2 25088 leibpi 25089 leibpisum 25090 wilthlem1 25214 wilthlem2 25215 basellem3 25229 zabsle1 25441 lgsvalmod 25461 lgsdir2lem4 25473 gausslemma2dlem6 25517 lgseisen 25524 ostth3 25747 axlowdimlem7 26254 ipidsq 28116 ipasslem10 28245 hisubcomi 28512 normlem9 28526 hmopd 29432 sgnclre 31143 sgnnbi 31149 sgnpbi 31150 sgnsgn 31152 signswch 31181 signstf 31186 signsvfn 31204 subfacval2 31711 iexpire 32159 bcneg1 32160 cnndvlem1 33055 ftc1anclem5 34031 asindmre 34037 dvasin 34038 dvacos 34039 dvreasin 34040 dvreacos 34041 areacirclem1 34042 stoweidlem22 41031 etransclem46 41289 smfneg 41802 3exp4mod41 42381 |
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