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Mirrors > Home > MPE Home > Th. List > logcnlem2 | Structured version Visualization version GIF version |
Description: Lemma for logcn 25233. (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 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ+) → 𝐴 ∈ ℝ+) | |
3 | logcnlem.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | logcn.d | . . . . . . . . . . 11 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
5 | 4 | ellogdm 25225 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 500 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | 3, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | 7 | imcld 14557 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
9 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℝ) |
10 | 9 | recnd 10672 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℂ) |
11 | reim0b 14481 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
13 | 5 | simprbi 499 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
14 | 3, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
15 | 12, 14 | sylbird 262 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ+)) |
16 | 15 | necon3bd 3033 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ ℝ+ → (ℑ‘𝐴) ≠ 0)) |
17 | 16 | imp 409 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ≠ 0) |
18 | 10, 17 | absrpcld 14811 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (abs‘(ℑ‘𝐴)) ∈ ℝ+) |
19 | 2, 18 | ifclda 4504 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ+) |
20 | 1, 19 | eqeltrid 2920 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ+) |
21 | logcnlem.t | . . 3 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) | |
22 | 4 | logdmn0 25226 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
23 | 3, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) |
24 | 7, 23 | absrpcld 14811 | . . . 4 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
25 | logcnlem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
26 | 1rp 12396 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
27 | rpaddcl 12414 | . . . . . 6 ⊢ ((1 ∈ ℝ+ ∧ 𝑅 ∈ ℝ+) → (1 + 𝑅) ∈ ℝ+) | |
28 | 26, 25, 27 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ+) |
29 | 25, 28 | rpdivcld 12451 | . . . 4 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ+) |
30 | 24, 29 | rpmulcld 12450 | . . 3 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ+) |
31 | 21, 30 | eqeltrid 2920 | . 2 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
32 | 20, 31 | ifcld 4515 | 1 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 ifcif 4470 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 -∞cmnf 10676 ≤ cle 10679 / cdiv 11300 ℝ+crp 12392 (,]cioc 12742 ℑcim 14460 abscabs 14596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ioc 12746 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 |
This theorem is referenced by: logcnlem5 25232 |
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