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Mirrors > Home > MPE Home > Th. List > logdmn0 | Structured version Visualization version GIF version |
Description: A number in the continuous domain of log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
logdmn0 | ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nrp 12412 | . . . 4 ⊢ ¬ 0 ∈ ℝ+ | |
2 | 0re 10632 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | logcn.d | . . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
4 | 3 | ellogdm 25230 | . . . . . 6 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ (0 ∈ ℝ → 0 ∈ ℝ+))) |
5 | 4 | simprbi 500 | . . . . 5 ⊢ (0 ∈ 𝐷 → (0 ∈ ℝ → 0 ∈ ℝ+)) |
6 | 2, 5 | mpi 20 | . . . 4 ⊢ (0 ∈ 𝐷 → 0 ∈ ℝ+) |
7 | 1, 6 | mto 200 | . . 3 ⊢ ¬ 0 ∈ 𝐷 |
8 | eleq1 2877 | . . 3 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐷 ↔ 0 ∈ 𝐷)) | |
9 | 7, 8 | mtbiri 330 | . 2 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐷) |
10 | 9 | necon2ai 3016 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 -∞cmnf 10662 ℝ+crp 12377 (,]cioc 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-addrcl 10587 ax-rnegex 10597 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-rp 12378 df-ioc 12731 |
This theorem is referenced by: logdmss 25233 logcnlem2 25234 logcnlem3 25235 logcnlem4 25236 logcnlem5 25237 logcn 25238 dvloglem 25239 logf1o2 25241 logtayl 25251 logtayl2 25253 dvcncxp1 25332 dvcnsqrt 25333 cxpcn 25334 atansssdm 25519 lgamgulmlem2 25615 |
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