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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 10331 | . 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 1483 ∈ wcel 1990 ℂcc 9934 -cneg 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 |
This theorem is referenced by: negn0 10459 ltnegcon1 10529 ltnegcon2 10530 lenegcon1 10532 lenegcon2 10533 negfi 10971 fiminre 10972 infm3lem 10981 infrenegsup 11006 zeo 11463 zindd 11478 znnn0nn 11489 supminf 11775 zsupss 11777 max0sub 12027 xnegneg 12045 ceilid 12650 expneg 12868 expaddzlem 12903 expaddz 12904 cjcj 13880 cnpart 13980 risefallfac 14755 sincossq 14906 bitsf1 15168 pcid 15577 4sqlem10 15651 mulgnegnn 17551 mulgsubcl 17555 mulgneg 17560 mulgz 17568 mulgass 17579 ghmmulg 17672 cyggeninv 18285 tgpmulg 21897 xrhmeo 22745 cphsqrtcl3 22987 iblneg 23569 itgneg 23570 ditgswap 23623 lhop2 23778 vieta1lem2 24066 ptolemy 24248 tanabsge 24258 tanord 24284 tanregt0 24285 lognegb 24336 logtayl 24406 logtayl2 24408 cxpmul2z 24437 isosctrlem2 24549 dcubic 24573 dquart 24580 atans2 24658 amgmlem 24716 lgamucov 24764 basellem5 24811 basellem9 24815 lgsdir2lem4 25053 dchrisum0flblem1 25197 ostth3 25327 ipasslem3 27688 ftc1anclem6 33490 rexzrexnn0 37368 acongsym 37543 acongneg2 37544 acongtr 37545 binomcxplemnotnn0 38555 infnsuprnmpt 39465 ltmulneg 39615 rexabslelem 39645 supminfrnmpt 39672 leneg2d 39676 leneg3d 39687 supminfxr 39694 climliminflimsupd 40033 itgsin0pilem1 40165 itgsinexplem1 40169 itgsincmulx 40190 stoweidlem13 40230 fourierdlem39 40363 fourierdlem43 40367 fourierdlem44 40368 etransclem46 40497 hoicvr 40762 smfinflem 41023 sigariz 41052 sigaradd 41055 amgmwlem 42548 |
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