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Mirrors > Home > MPE Home > Th. List > logdmss | Structured version Visualization version GIF version |
Description: The continuity domain of log is a subset of the regular domain of log. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
Ref | Expression |
---|---|
logdmss | ⊢ 𝐷 ⊆ (ℂ ∖ {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcn.d | . . . . 5 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
2 | 1 | ellogdm 26696 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
3 | 2 | simplbi 497 | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
4 | 1 | logdmn0 26697 | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
5 | eldifsn 4791 | . . 3 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
6 | 3, 4, 5 | sylanbrc 583 | . 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ (ℂ ∖ {0})) |
7 | 6 | ssriv 3999 | 1 ⊢ 𝐷 ⊆ (ℂ ∖ {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 -∞cmnf 11291 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-addrcl 11214 ax-rnegex 11224 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
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-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-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-rp 13033 df-ioc 13389 |
This theorem is referenced by: logcn 26704 dvloglem 26705 logf1o2 26707 dvlog 26708 dvlog2 26710 logtayl 26717 dvatan 26993 efrlim 27027 efrlimOLD 27028 lgamcvg2 27113 |
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