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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 10631 | . . 3 ⊢ 0 ∈ ℝ | |
2 | lesub2 11123 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) | |
3 | 1, 2 | mp3an3 1441 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) |
4 | df-neg 10861 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
5 | df-neg 10861 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
6 | 4, 5 | breq12i 5066 | . 2 ⊢ (-𝐵 ≤ -𝐴 ↔ (0 − 𝐵) ≤ (0 − 𝐴)) |
7 | 3, 6 | syl6bbr 290 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 0cc0 10525 ≤ cle 10664 − cmin 10858 -cneg 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 |
This theorem is referenced by: lenegcon1 11132 lenegcon2 11133 le0neg1 11136 le0neg2 11137 leord2 11158 lenegi 11173 lenegd 11207 infm3 11588 uzneg 12251 zmax 12333 rebtwnz 12335 iccneg 12846 aaliou3lem2 24859 logreclem 25267 atanlogsublem 25420 emcllem7 25506 ltflcei 34761 smfinflem 42968 |
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