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Mirrors > Home > MPE Home > Th. List > logcnlem2 | Structured version Visualization version GIF version |
Description: Lemma for logcn 24592. (Contributed by Mario Carneiro, 25-Feb-2015.) |
Ref | Expression |
---|---|
logcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
logcnlem.s | ⊢ 𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) |
logcnlem.t | ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) |
logcnlem.a | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
logcnlem.r | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
Ref | Expression |
---|---|
logcnlem2 | ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logcnlem.s | . . 3 ⊢ 𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) | |
2 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
3 | logcnlem.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | logcn.d | . . . . . . . . . . 11 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
5 | 4 | ellogdm 24584 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 478 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | 3, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | 7 | imcld 14134 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
9 | 8 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℝ) |
10 | 9 | recnd 10260 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℂ) |
11 | reim0b 14058 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
13 | 5 | simprbi 483 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
14 | 3, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
15 | 12, 14 | sylbird 250 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ+)) |
16 | 15 | necon3bd 2946 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ ℝ+ → (ℑ‘𝐴) ≠ 0)) |
17 | 16 | imp 444 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ≠ 0) |
18 | 10, 17 | absrpcld 14386 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (abs‘(ℑ‘𝐴)) ∈ ℝ+) |
19 | 2, 18 | ifclda 4264 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ+) |
20 | 1, 19 | syl5eqel 2843 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ+) |
21 | logcnlem.t | . . 3 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) | |
22 | 4 | logdmn0 24585 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
23 | 3, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) |
24 | 7, 23 | absrpcld 14386 | . . . 4 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
25 | logcnlem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
26 | 1rp 12029 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
27 | rpaddcl 12047 | . . . . . 6 ⊢ ((1 ∈ ℝ+ ∧ 𝑅 ∈ ℝ+) → (1 + 𝑅) ∈ ℝ+) | |
28 | 26, 25, 27 | sylancr 698 | . . . . 5 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ+) |
29 | 25, 28 | rpdivcld 12082 | . . . 4 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ+) |
30 | 24, 29 | rpmulcld 12081 | . . 3 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ+) |
31 | 21, 30 | syl5eqel 2843 | . 2 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
32 | 20, 31 | ifcld 4275 | 1 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∖ cdif 3712 ifcif 4230 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 ℝcr 10127 0cc0 10128 1c1 10129 + caddc 10131 · cmul 10133 -∞cmnf 10264 ≤ cle 10267 / cdiv 10876 ℝ+crp 12025 (,]cioc 12369 ℑcim 14037 abscabs 14173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-ioc 12373 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 |
This theorem is referenced by: logcnlem5 24591 |
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