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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 12427 | . . . 4 ⊢ ¬ 0 ∈ ℝ+ | |
2 | 0re 10645 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | logcn.d | . . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
4 | 3 | ellogdm 25224 | . . . . . 6 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ (0 ∈ ℝ → 0 ∈ ℝ+))) |
5 | 4 | simprbi 499 | . . . . 5 ⊢ (0 ∈ 𝐷 → (0 ∈ ℝ → 0 ∈ ℝ+)) |
6 | 2, 5 | mpi 20 | . . . 4 ⊢ (0 ∈ 𝐷 → 0 ∈ ℝ+) |
7 | 1, 6 | mto 199 | . . 3 ⊢ ¬ 0 ∈ 𝐷 |
8 | eleq1 2902 | . . 3 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐷 ↔ 0 ∈ 𝐷)) | |
9 | 7, 8 | mtbiri 329 | . 2 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐷) |
10 | 9 | necon2ai 3047 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 -∞cmnf 10675 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 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-rp 12393 df-ioc 12746 |
This theorem is referenced by: logdmss 25227 logcnlem2 25228 logcnlem3 25229 logcnlem4 25230 logcnlem5 25231 logcn 25232 dvloglem 25233 logf1o2 25235 logtayl 25245 logtayl2 25247 dvcncxp1 25326 dvcnsqrt 25327 cxpcn 25328 atansssdm 25513 lgamgulmlem2 25609 |
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