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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 3695 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝐴 ∈ (-∞(,]0)) | |
2 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 1, 2 | eleq2s 2706 | . 2 ⊢ (𝐴 ∈ 𝐷 → ¬ 𝐴 ∈ (-∞(,]0)) |
4 | rpre 11715 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
5 | 2 | ellogdm 24185 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 475 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | negreb 10225 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
9 | 4, 8 | syl5ib 233 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)) |
10 | 9 | imp 444 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
11 | mnflt 11833 | . . . 4 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
12 | 10, 11 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → -∞ < 𝐴) |
13 | rpgt0 11720 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ+ → 0 < -𝐴) | |
14 | 13 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 0 < -𝐴) |
15 | 10 | lt0neg1d 10476 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 ↔ 0 < -𝐴)) |
16 | 14, 15 | mpbird 246 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 < 0) |
17 | 0re 9919 | . . . . 5 ⊢ 0 ∈ ℝ | |
18 | ltle 10005 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0)) | |
19 | 10, 17, 18 | sylancl 693 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 → 𝐴 ≤ 0)) |
20 | 16, 19 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ≤ 0) |
21 | mnfxr 9975 | . . . 4 ⊢ -∞ ∈ ℝ* | |
22 | elioc2 12107 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))) | |
23 | 21, 17, 22 | mp2an 704 | . . 3 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)) |
24 | 10, 12, 20, 23 | syl3anbrc 1239 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ (-∞(,]0)) |
25 | 3, 24 | mtand 689 | 1 ⊢ (𝐴 ∈ 𝐷 → ¬ -𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 -cneg 10146 ℝ+crp 11708 (,]cioc 12047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-rp 11709 df-ioc 12051 |
This theorem is referenced by: dvloglem 24194 logf1o2 24196 |
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