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Mirrors > Home > MPE Home > Th. List > logdmnrp | Structured version Visualization version GIF version |
Description: A number in the continuous domain of log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
logdmnrp | ⊢ (𝐴 ∈ 𝐷 → ¬ -𝐴 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifn 4028 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝐴 ∈ (-∞(,]0)) | |
2 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 1, 2 | eleq2s 2852 | . 2 ⊢ (𝐴 ∈ 𝐷 → ¬ 𝐴 ∈ (-∞(,]0)) |
4 | rpre 12492 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
5 | 2 | ellogdm 25394 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 501 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | negreb 11041 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
9 | 4, 8 | syl5ib 247 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)) |
10 | 9 | imp 410 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
11 | 10 | mnfltd 12614 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → -∞ < 𝐴) |
12 | rpgt0 12496 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ+ → 0 < -𝐴) | |
13 | 12 | adantl 485 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 0 < -𝐴) |
14 | 10 | lt0neg1d 11299 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 ↔ 0 < -𝐴)) |
15 | 13, 14 | mpbird 260 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 < 0) |
16 | 0re 10733 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | ltle 10819 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0)) | |
18 | 10, 16, 17 | sylancl 589 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 → 𝐴 ≤ 0)) |
19 | 15, 18 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ≤ 0) |
20 | mnfxr 10788 | . . . 4 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 12896 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))) | |
22 | 20, 16, 21 | mp2an 692 | . . 3 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)) |
23 | 10, 11, 19, 22 | syl3anbrc 1344 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ (-∞(,]0)) |
24 | 3, 23 | mtand 816 | 1 ⊢ (𝐴 ∈ 𝐷 → ¬ -𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∖ cdif 3850 class class class wbr 5040 (class class class)co 7182 ℂcc 10625 ℝcr 10626 0cc0 10627 -∞cmnf 10763 ℝ*cxr 10764 < clt 10765 ≤ cle 10766 -cneg 10961 ℝ+crp 12484 (,]cioc 12834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-rp 12485 df-ioc 12838 |
This theorem is referenced by: dvloglem 25403 logf1o2 25405 |
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