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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 11741 | . . . 4 ⊢ ¬ 0 ∈ ℝ+ | |
2 | 0re 9919 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | logcn.d | . . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
4 | 3 | ellogdm 24185 | . . . . . 6 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ (0 ∈ ℝ → 0 ∈ ℝ+))) |
5 | 4 | simprbi 479 | . . . . 5 ⊢ (0 ∈ 𝐷 → (0 ∈ ℝ → 0 ∈ ℝ+)) |
6 | 2, 5 | mpi 20 | . . . 4 ⊢ (0 ∈ 𝐷 → 0 ∈ ℝ+) |
7 | 1, 6 | mto 187 | . . 3 ⊢ ¬ 0 ∈ 𝐷 |
8 | eleq1 2676 | . . 3 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐷 ↔ 0 ∈ 𝐷)) | |
9 | 7, 8 | mtbiri 316 | . 2 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐷) |
10 | 9 | necon2ai 2811 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 -∞cmnf 9951 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-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
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-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-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-rp 11709 df-ioc 12051 |
This theorem is referenced by: logdmss 24188 logcnlem2 24189 logcnlem3 24190 logcnlem4 24191 logcnlem5 24192 logcn 24193 dvloglem 24194 logf1o2 24196 logtayl 24206 logtayl2 24208 dvcncxp1 24284 dvcnsqrt 24285 cxpcn 24286 atansssdm 24460 lgamgulmlem2 24556 |
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