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Mirrors > Home > MPE Home > Th. List > lenegcon1 | Structured version Visualization version GIF version |
Description: Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
lenegcon1 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl 11553 | . . 3 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
2 | leneg 11747 | . . 3 ⊢ ((-𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ --𝐴)) | |
3 | 1, 2 | sylan 578 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ --𝐴)) |
4 | recn 11228 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
5 | 4 | negnegd 11592 | . . . 4 ⊢ (𝐴 ∈ ℝ → --𝐴 = 𝐴) |
6 | 5 | breq2d 5155 | . . 3 ⊢ (𝐴 ∈ ℝ → (-𝐵 ≤ --𝐴 ↔ -𝐵 ≤ 𝐴)) |
7 | 6 | adantr 479 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐵 ≤ --𝐴 ↔ -𝐵 ≤ 𝐴)) |
8 | 3, 7 | bitrd 278 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5143 ℝcr 11137 ≤ cle 11279 -cneg 11475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 |
This theorem is referenced by: lenegcon1i 11796 lenegcon1d 11826 ublbneg 12947 zmax 12959 absle 15294 lenegsq 15299 abs2difabs 15313 o1lo1 15513 infcvgaux2i 15836 sinbnd 16156 cosbnd 16157 xrhmeo 24889 logcj 26558 asinneg 26836 |
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