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Mirrors > Home > MPE Home > Th. List > lt0neg1 | Structured version Visualization version GIF version |
Description: Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.) |
Ref | Expression |
---|---|
lt0neg1 | ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10365 | . . 3 ⊢ 0 ∈ ℝ | |
2 | ltneg 10859 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 ↔ -0 < -𝐴)) | |
3 | 1, 2 | mpan2 682 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ -0 < -𝐴)) |
4 | neg0 10655 | . . 3 ⊢ -0 = 0 | |
5 | 4 | breq1i 4882 | . 2 ⊢ (-0 < -𝐴 ↔ 0 < -𝐴) |
6 | 3, 5 | syl6bb 279 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 class class class wbr 4875 ℝcr 10258 0cc0 10259 < clt 10398 -cneg 10593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 |
This theorem is referenced by: mullt0 10878 lt0neg1d 10928 recgt0ii 11266 rpneg 12153 negelrp 12154 divalglem6 15502 sinhalfpilem 24622 atanbnd 25073 sgnmulsgp 31154 mulltgt0 39998 ztprmneprm 42990 |
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