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Mirrors > Home > MPE Home > Th. List > negnegd | Structured version Visualization version GIF version |
Description: A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
negnegd | ⊢ (𝜑 → --𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | negneg 11560 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → --𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ℂcc 11156 -cneg 11495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-neg 11497 |
This theorem is referenced by: negn0 11693 ltnegcon1 11765 ltnegcon2 11766 lenegcon1 11768 lenegcon2 11769 negfi 12215 infm3lem 12224 infrenegsup 12249 zeo 12700 zindd 12715 znnn0nn 12725 supminf 12971 zsupss 12973 max0sub 13229 xnegneg 13247 ceilid 13871 expneg 14089 expaddzlem 14125 expaddz 14126 cjcj 15145 cnpart 15245 risefallfac 16026 sincossq 16178 bitsf1 16446 pcid 16875 4sqlem10 16949 mulgnegnn 19078 mulgsubcl 19082 mulgneg 19086 mulgz 19096 mulgass 19105 ghmmulg 19222 cyggeninv 19881 tgpmulg 24088 xrhmeo 24962 cphsqrtcl3 25206 iblneg 25823 itgneg 25824 ditgswap 25879 lhop2 26039 vieta1lem2 26339 ptolemy 26524 tanabsge 26534 tanord 26565 tanregt0 26566 lognegb 26617 logtayl 26687 logtayl2 26689 cxpmul2z 26718 isosctrlem2 26847 dcubic 26874 dquart 26881 atans2 26959 amgmlem 27018 lgamucov 27066 basellem5 27113 basellem9 27117 lgsdir2lem4 27357 dchrisum0flblem1 27537 ostth3 27667 ipasslem3 30766 ftc1anclem6 37399 lcmineqlem12 41739 posbezout 41798 dffltz 42288 rexzrexnn0 42461 acongsym 42634 acongneg2 42635 acongtr 42636 binomcxplemnotnn0 44030 infnsuprnmpt 44859 ltmulneg 45007 rexabslelem 45033 supminfrnmpt 45060 leneg2d 45063 leneg3d 45072 supminfxr 45079 climliminflimsupd 45422 itgsin0pilem1 45571 itgsinexplem1 45575 itgsincmulx 45595 stoweidlem13 45634 fourierdlem39 45767 fourierdlem43 45771 fourierdlem44 45772 etransclem46 45901 hoicvr 46169 smfinflem 46438 sigariz 46484 sigaradd 46487 sqrtnegnre 46920 requad01 47193 itsclc0yqsol 48152 amgmwlem 48550 |
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