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Mirrors > Home > MPE Home > Th. List > logcnlem2 | Structured version Visualization version GIF version |
Description: Lemma for logcn 26668. (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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
3 | logcnlem.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | logcn.d | . . . . . . . . . . 11 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
5 | 4 | ellogdm 26660 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 496 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | 3, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | 7 | imcld 15192 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
9 | 8 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℝ) |
10 | 9 | recnd 11280 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℂ) |
11 | reim0b 15116 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
13 | 5 | simprbi 495 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
14 | 3, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
15 | 12, 14 | sylbird 259 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ+)) |
16 | 15 | necon3bd 2944 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ ℝ+ → (ℑ‘𝐴) ≠ 0)) |
17 | 16 | imp 405 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ≠ 0) |
18 | 10, 17 | absrpcld 15445 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (abs‘(ℑ‘𝐴)) ∈ ℝ+) |
19 | 2, 18 | ifclda 4558 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ+) |
20 | 1, 19 | eqeltrid 2830 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ+) |
21 | logcnlem.t | . . 3 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) | |
22 | 4 | logdmn0 26661 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
23 | 3, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) |
24 | 7, 23 | absrpcld 15445 | . . . 4 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
25 | logcnlem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
26 | 1rp 13023 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
27 | rpaddcl 13041 | . . . . . 6 ⊢ ((1 ∈ ℝ+ ∧ 𝑅 ∈ ℝ+) → (1 + 𝑅) ∈ ℝ+) | |
28 | 26, 25, 27 | sylancr 585 | . . . . 5 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ+) |
29 | 25, 28 | rpdivcld 13078 | . . . 4 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ+) |
30 | 24, 29 | rpmulcld 13077 | . . 3 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ+) |
31 | 21, 30 | eqeltrid 2830 | . 2 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
32 | 20, 31 | ifcld 4569 | 1 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3943 ifcif 4523 class class class wbr 5143 ‘cfv 6543 (class class class)co 7413 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 -∞cmnf 11284 ≤ cle 11287 / cdiv 11909 ℝ+crp 13019 (,]cioc 13370 ℑcim 15095 abscabs 15231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9475 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-n0 12516 df-z 12602 df-uz 12866 df-rp 13020 df-ioc 13374 df-seq 14013 df-exp 14073 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 |
This theorem is referenced by: logcnlem5 26667 |
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