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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 25216 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
3 | 2 | simplbi 500 | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
4 | 1 | logdmn0 25217 | . . 3 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
5 | eldifsn 4712 | . . 3 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
6 | 3, 4, 5 | sylanbrc 585 | . 2 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ (ℂ ∖ {0})) |
7 | 6 | ssriv 3970 | 1 ⊢ 𝐷 ⊆ (ℂ ∖ {0}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 -∞cmnf 10667 ℝ+crp 12383 (,]cioc 12733 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-addrcl 10592 ax-rnegex 10602 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-rp 12384 df-ioc 12737 |
This theorem is referenced by: logcn 25224 dvloglem 25225 logf1o2 25227 dvlog 25228 dvlog2 25230 logtayl 25237 dvatan 25507 efrlim 25541 lgamcvg2 25626 |
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