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| 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 1539 ∈ wcel 2107 ℂcc 11154 -cneg 11494 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: negn0 11693 ltnegcon1 11765 ltnegcon2 11766 lenegcon1 11768 lenegcon2 11769 negfi 12218 infm3lem 12227 infrenegsup 12252 zeo 12706 zindd 12721 znnn0nn 12731 supminf 12978 zsupss 12980 max0sub 13239 xnegneg 13257 ceilid 13892 expneg 14111 expaddzlem 14147 expaddz 14148 cjcj 15180 cnpart 15280 risefallfac 16061 sincossq 16213 bitsf1 16484 pcid 16912 4sqlem10 16986 mulgnegnn 19103 mulgsubcl 19107 mulgneg 19111 mulgz 19121 mulgass 19130 ghmmulg 19247 cyggeninv 19902 tgpmulg 24102 xrhmeo 24978 cphsqrtcl3 25222 iblneg 25839 itgneg 25840 ditgswap 25895 lhop2 26055 vieta1lem2 26354 ptolemy 26539 tanabsge 26549 tanord 26581 tanregt0 26582 lognegb 26633 logtayl 26703 logtayl2 26705 cxpmul2z 26734 isosctrlem2 26863 dcubic 26890 dquart 26897 atans2 26975 amgmlem 27034 lgamucov 27082 basellem5 27129 basellem9 27133 lgsdir2lem4 27373 dchrisum0flblem1 27553 ostth3 27683 ipasslem3 30853 zrhcntr 33981 ftc1anclem6 37706 lcmineqlem12 42042 posbezout 42102 dffltz 42649 rexzrexnn0 42820 acongsym 42993 acongneg2 42994 acongtr 42995 binomcxplemnotnn0 44380 infnsuprnmpt 45262 ltmulneg 45408 rexabslelem 45434 supminfrnmpt 45461 leneg2d 45464 leneg3d 45473 supminfxr 45480 climliminflimsupd 45821 itgsin0pilem1 45970 itgsinexplem1 45974 itgsincmulx 45994 stoweidlem13 46033 fourierdlem39 46166 fourierdlem43 46170 fourierdlem44 46171 etransclem46 46300 hoicvr 46568 smfinflem 46837 sigariz 46883 sigaradd 46886 sqrtnegnre 47324 ceildivmod 47346 requad01 47613 itsclc0yqsol 48690 amgmwlem 49376 | 
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