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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 4107 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝐴 ∈ (-∞(,]0)) | |
2 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 1, 2 | eleq2s 2934 | . 2 ⊢ (𝐴 ∈ 𝐷 → ¬ 𝐴 ∈ (-∞(,]0)) |
4 | rpre 12400 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
5 | 2 | ellogdm 25225 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 500 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | negreb 10954 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
9 | 4, 8 | syl5ib 246 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)) |
10 | 9 | imp 409 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
11 | 10 | mnfltd 12522 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → -∞ < 𝐴) |
12 | rpgt0 12404 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ+ → 0 < -𝐴) | |
13 | 12 | adantl 484 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 0 < -𝐴) |
14 | 10 | lt0neg1d 11212 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 ↔ 0 < -𝐴)) |
15 | 13, 14 | mpbird 259 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 < 0) |
16 | 0re 10646 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | ltle 10732 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0)) | |
18 | 10, 16, 17 | sylancl 588 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 → 𝐴 ≤ 0)) |
19 | 15, 18 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ≤ 0) |
20 | mnfxr 10701 | . . . 4 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 12802 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))) | |
22 | 20, 16, 21 | mp2an 690 | . . 3 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)) |
23 | 10, 11, 19, 22 | syl3anbrc 1339 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ (-∞(,]0)) |
24 | 3, 23 | mtand 814 | 1 ⊢ (𝐴 ∈ 𝐷 → ¬ -𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∖ cdif 3936 class class class wbr 5069 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 -∞cmnf 10676 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 -cneg 10874 ℝ+crp 12392 (,]cioc 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-rp 12393 df-ioc 12746 |
This theorem is referenced by: dvloglem 25234 logf1o2 25236 |
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