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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 10629 | . 2 ⊢ 1 ∈ ℝ | |
2 | 1 | renegcli 10935 | 1 ⊢ -1 ∈ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ℝcr 10524 1c1 10526 -cneg 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-sub 10860 df-neg 10861 |
This theorem is referenced by: dfceil2 13197 bernneq 13578 crre 14461 remim 14464 iseraltlem2 15027 iseraltlem3 15028 iseralt 15029 tanhbnd 15502 sinbnd2 15523 cosbnd2 15524 psgnodpmr 20662 xrhmeo 23477 xrhmph 23478 vitalilem2 24137 vitalilem4 24139 vitali 24141 mbfneg 24178 i1fsub 24236 itg1sub 24237 i1fibl 24335 itgitg1 24336 recosf1o 25046 efif1olem3 25055 relogbdiv 25284 ang180lem3 25316 1cubrlem 25346 atanre 25390 acosrecl 25408 atandmcj 25414 leibpilem2 25446 leibpi 25447 leibpisum 25448 wilthlem1 25572 wilthlem2 25573 basellem3 25587 zabsle1 25799 lgsvalmod 25819 lgsdir2lem4 25831 gausslemma2dlem6 25875 lgseisen 25882 ostth3 26141 axlowdimlem7 26661 ipidsq 28414 ipasslem10 28543 hisubcomi 28808 normlem9 28822 hmopd 29726 sgnclre 31696 sgnnbi 31702 sgnpbi 31703 sgnsgn 31705 signswch 31730 signstf 31735 signsvfn 31751 subfacval2 32331 iexpire 32864 bcneg1 32865 cnndvlem1 33773 ftc1anclem5 34852 asindmre 34858 dvasin 34859 dvacos 34860 dvreasin 34861 dvreacos 34862 areacirclem1 34863 stoweidlem22 42184 etransclem46 42442 smfneg 42955 3exp4mod41 43658 |
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