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Mirrors > Home > MPE Home > Th. List > logcnlem2 | Structured version Visualization version GIF version |
Description: Lemma for logcn 24793. (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 24785 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
6 | 5 | simplbi 493 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ ℂ) |
7 | 3, 6 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
8 | 7 | imcld 14313 | . . . . . . 7 ⊢ (𝜑 → (ℑ‘𝐴) ∈ ℝ) |
9 | 8 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℝ) |
10 | 9 | recnd 10386 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ∈ ℂ) |
11 | reim0b 14237 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0)) |
13 | 5 | simprbi 492 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝐷 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
14 | 3, 13 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
15 | 12, 14 | sylbird 252 | . . . . . . 7 ⊢ (𝜑 → ((ℑ‘𝐴) = 0 → 𝐴 ∈ ℝ+)) |
16 | 15 | necon3bd 3014 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 ∈ ℝ+ → (ℑ‘𝐴) ≠ 0)) |
17 | 16 | imp 397 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (ℑ‘𝐴) ≠ 0) |
18 | 10, 17 | absrpcld 14565 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ∈ ℝ+) → (abs‘(ℑ‘𝐴)) ∈ ℝ+) |
19 | 2, 18 | ifclda 4341 | . . 3 ⊢ (𝜑 → if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴))) ∈ ℝ+) |
20 | 1, 19 | syl5eqel 2911 | . 2 ⊢ (𝜑 → 𝑆 ∈ ℝ+) |
21 | logcnlem.t | . . 3 ⊢ 𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) | |
22 | 4 | logdmn0 24786 | . . . . . 6 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ≠ 0) |
23 | 3, 22 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 0) |
24 | 7, 23 | absrpcld 14565 | . . . 4 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
25 | logcnlem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
26 | 1rp 12117 | . . . . . 6 ⊢ 1 ∈ ℝ+ | |
27 | rpaddcl 12137 | . . . . . 6 ⊢ ((1 ∈ ℝ+ ∧ 𝑅 ∈ ℝ+) → (1 + 𝑅) ∈ ℝ+) | |
28 | 26, 25, 27 | sylancr 583 | . . . . 5 ⊢ (𝜑 → (1 + 𝑅) ∈ ℝ+) |
29 | 25, 28 | rpdivcld 12174 | . . . 4 ⊢ (𝜑 → (𝑅 / (1 + 𝑅)) ∈ ℝ+) |
30 | 24, 29 | rpmulcld 12173 | . . 3 ⊢ (𝜑 → ((abs‘𝐴) · (𝑅 / (1 + 𝑅))) ∈ ℝ+) |
31 | 21, 30 | syl5eqel 2911 | . 2 ⊢ (𝜑 → 𝑇 ∈ ℝ+) |
32 | 20, 31 | ifcld 4352 | 1 ⊢ (𝜑 → if(𝑆 ≤ 𝑇, 𝑆, 𝑇) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∖ cdif 3796 ifcif 4307 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 ℝcr 10252 0cc0 10253 1c1 10254 + caddc 10256 · cmul 10258 -∞cmnf 10390 ≤ cle 10393 / cdiv 11010 ℝ+crp 12113 (,]cioc 12465 ℑcim 14216 abscabs 14352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-ioc 12469 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 |
This theorem is referenced by: logcnlem5 24792 |
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