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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 11203 | . . 3 ⊢ 0 ∈ ℝ | |
2 | lesub2 11696 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) | |
3 | 1, 2 | mp3an3 1451 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (0 − 𝐵) ≤ (0 − 𝐴))) |
4 | df-neg 11434 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
5 | df-neg 11434 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
6 | 4, 5 | breq12i 5153 | . 2 ⊢ (-𝐵 ≤ -𝐴 ↔ (0 − 𝐵) ≤ (0 − 𝐴)) |
7 | 3, 6 | bitr4di 289 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 class class class wbr 5144 (class class class)co 7396 ℝcr 11096 0cc0 11097 ≤ cle 11236 − cmin 11431 -cneg 11432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 |
This theorem is referenced by: lenegcon1 11705 lenegcon2 11706 le0neg1 11709 le0neg2 11710 leord2 11731 lenegi 11746 lenegd 11780 infm3 12160 uzneg 12829 zmax 12916 rebtwnz 12918 iccneg 13436 aaliou3lem2 25825 logreclem 26234 atanlogsublem 26387 emcllem7 26473 ltflcei 36381 smfinflem 45406 |
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