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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 4142 | . . 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 13041 | . . . . 5 ⊢ (-𝐴 ∈ ℝ+ → -𝐴 ∈ ℝ) | |
5 | 2 | ellogdm 26696 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 497 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | negreb 11572 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ)) |
9 | 4, 8 | imbitrid 244 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (-𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ)) |
10 | 9 | imp 406 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ) |
11 | 10 | mnfltd 13164 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → -∞ < 𝐴) |
12 | rpgt0 13045 | . . . . . 6 ⊢ (-𝐴 ∈ ℝ+ → 0 < -𝐴) | |
13 | 12 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 0 < -𝐴) |
14 | 10 | lt0neg1d 11830 | . . . . 5 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → (𝐴 < 0 ↔ 0 < -𝐴)) |
15 | 13, 14 | mpbird 257 | . . . 4 ⊢ ((𝐴 ∈ 𝐷 ∧ -𝐴 ∈ ℝ+) → 𝐴 < 0) |
16 | 0re 11261 | . . . . 5 ⊢ 0 ∈ ℝ | |
17 | ltle 11347 | . . . . 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 11316 | . . . 4 ⊢ -∞ ∈ ℝ* | |
21 | elioc2 13447 | . . . 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 1342 | . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 -cneg 11491 ℝ+crp 13032 (,]cioc 13385 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-rp 13033 df-ioc 13389 |
This theorem is referenced by: dvloglem 26705 logf1o2 26707 |
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