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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 4062 | . . 3 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → ¬ 𝐴 ∈ (-∞(,]0)) | |
2 | logcn.d | . . 3 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
3 | 1, 2 | eleq2s 2857 | . 2 ⊢ (𝐴 ∈ 𝐷 → ¬ 𝐴 ∈ (-∞(,]0)) |
4 | rpre 12738 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
5 | 2 | ellogdm 25794 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 498 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | negreb 11286 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
9 | 4, 8 | syl5ib 243 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)) |
10 | 9 | imp 407 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
11 | 10 | mnfltd 12860 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → -∞ < 𝐴) |
12 | rpgt0 12742 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ+ → 0 < -𝐴) | |
13 | 12 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 0 < -𝐴) |
14 | 10 | lt0neg1d 11544 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 ↔ 0 < -𝐴)) |
15 | 13, 14 | mpbird 256 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 < 0) |
16 | 0re 10977 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | ltle 11063 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 < 0 → 𝐴 ≤ 0)) | |
18 | 10, 16, 17 | sylancl 586 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 → 𝐴 ≤ 0)) |
19 | 15, 18 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ≤ 0) |
20 | mnfxr 11032 | . . . 4 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 13142 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ) → (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0))) | |
22 | 20, 16, 21 | mp2an 689 | . . 3 ⊢ (𝐴 ∈ (-∞(,]0) ↔ (𝐴 ∈ ℝ ∧ -∞ < 𝐴 ∧ 𝐴 ≤ 0)) |
23 | 10, 11, 19, 22 | syl3anbrc 1342 | . 2 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ (-∞(,]0)) |
24 | 3, 23 | mtand 813 | 1 ⊢ (𝐴 ∈ 𝐷 → ¬ -𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 class class class wbr 5074 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 -∞cmnf 11007 ℝ*cxr 11008 < clt 11009 ≤ cle 11010 -cneg 11206 ℝ+crp 12730 (,]cioc 13080 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-rp 12731 df-ioc 13084 |
This theorem is referenced by: dvloglem 25803 logf1o2 25805 |
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