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Mirrors > Home > MPE Home > Th. List > xle0neg1 | Structured version Visualization version GIF version |
Description: Extended real version of le0neg1 11147. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
xle0neg1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10687 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xleneg 12610 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ≤ 0 ↔ -𝑒0 ≤ -𝑒𝐴)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ -𝑒0 ≤ -𝑒𝐴)) |
4 | xneg0 12604 | . . 3 ⊢ -𝑒0 = 0 | |
5 | 4 | breq1i 5072 | . 2 ⊢ (-𝑒0 ≤ -𝑒𝐴 ↔ 0 ≤ -𝑒𝐴) |
6 | 3, 5 | syl6bb 289 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ 0 ↔ 0 ≤ -𝑒𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5065 0cc0 10536 ℝ*cxr 10673 ≤ cle 10675 -𝑒cxne 12503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-xneg 12506 |
This theorem is referenced by: xrhmeo 23549 |
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