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Mirrors > Home > MPE Home > Th. List > negneg | 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 NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negneg | ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 11346 | . . 3 ⊢ --𝐴 = (0 − -𝐴) | |
2 | 0cn 11105 | . . . 4 ⊢ 0 ∈ ℂ | |
3 | subneg 11408 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 − -𝐴) = (0 + 𝐴)) | |
4 | 2, 3 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℂ → (0 − -𝐴) = (0 + 𝐴)) |
5 | 1, 4 | eqtrid 2789 | . 2 ⊢ (𝐴 ∈ ℂ → --𝐴 = (0 + 𝐴)) |
6 | addid2 11296 | . 2 ⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | |
7 | 5, 6 | eqtrd 2777 | 1 ⊢ (𝐴 ∈ ℂ → --𝐴 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 (class class class)co 7351 ℂcc 11007 0cc0 11009 + caddc 11012 − cmin 11343 -cneg 11344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-sub 11345 df-neg 11346 |
This theorem is referenced by: neg11 11410 negcon1 11411 negreb 11424 negnegi 11429 negnegd 11461 negf1o 11543 mul2neg 11552 divneg2 11837 negfi 12062 nnnegz 12460 znegclb 12498 expneg2 13930 shftcan2 14928 sqreulem 15203 sqreu 15204 fallrisefac 15867 dvdsnegb 16115 lognegb 25896 logcj 25912 argimgt0 25918 cxpsqrt 26009 eldmgm 26322 supminfxr 43597 |
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