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Mirrors > Home > MPE Home > Th. List > leneg | Structured version Visualization version GIF version |
Description: Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leneg | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10242 | . . 3 ⊢ 0 ∈ ℝ | |
2 | lesub2 10725 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) | |
3 | 1, 2 | mp3an3 1561 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) |
4 | df-neg 10471 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
5 | df-neg 10471 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
6 | 4, 5 | breq12i 4795 | . 2 ⊢ (-𝐵 ≤ -𝐴 ↔ (0 − 𝐵) ≤ (0 − 𝐴)) |
7 | 3, 6 | syl6bbr 278 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6793 ℝcr 10137 0cc0 10138 ≤ cle 10277 − cmin 10468 -cneg 10469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 |
This theorem is referenced by: lenegcon1 10734 lenegcon2 10735 le0neg1 10738 le0neg2 10739 leord2 10760 lenegi 10775 lenegd 10808 infm3 11184 uzneg 11907 zmax 11988 rebtwnz 11990 iccneg 12500 aaliou3lem2 24318 logreclem 24721 atanlogsublem 24863 emcllem7 24949 ltflcei 33730 smfinflem 41543 |
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