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Mirrors > Home > MPE Home > Th. List > logcnlem2 | Structured version Visualization version GIF version |
Description: Lemma for logcn 25238. (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 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
3 | logcnlem.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | logcn.d | . . . . . . . . . . 11 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
5 | 4 | ellogdm 25230 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 501 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | 3, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | 7 | imcld 14546 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
9 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℝ) |
10 | 9 | recnd 10658 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℂ) |
11 | reim0b 14470 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
13 | 5 | simprbi 500 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
14 | 3, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
15 | 12, 14 | sylbird 263 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ+)) |
16 | 15 | necon3bd 3001 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ ℝ+ → (ℑ‘𝐴) ≠ 0)) |
17 | 16 | imp 410 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ≠ 0) |
18 | 10, 17 | absrpcld 14800 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (abs‘(ℑ‘𝐴)) ∈ ℝ+) |
19 | 2, 18 | ifclda 4459 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ+) |
20 | 1, 19 | eqeltrid 2894 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ+) |
21 | logcnlem.t | . . 3 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) | |
22 | 4 | logdmn0 25231 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
23 | 3, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) |
24 | 7, 23 | absrpcld 14800 | . . . 4 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
25 | logcnlem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
26 | 1rp 12381 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
27 | rpaddcl 12399 | . . . . . 6 ⊢ ((1 ∈ ℝ+ ∧ 𝑅 ∈ ℝ+) → (1 + 𝑅) ∈ ℝ+) | |
28 | 26, 25, 27 | sylancr 590 | . . . . 5 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ+) |
29 | 25, 28 | rpdivcld 12436 | . . . 4 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ+) |
30 | 24, 29 | rpmulcld 12435 | . . 3 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ+) |
31 | 21, 30 | eqeltrid 2894 | . 2 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
32 | 20, 31 | ifcld 4470 | 1 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ifcif 4425 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 -∞cmnf 10662 ≤ cle 10665 / cdiv 11286 ℝ+crp 12377 (,]cioc 12727 ℑcim 14449 abscabs 14585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ioc 12731 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 |
This theorem is referenced by: logcnlem5 25237 |
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