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Mirrors > Home > MPE Home > Th. List > Mathboxes > isosctrlem1ALT | Structured version Visualization version GIF version |
Description: Lemma for isosctr 25393. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 25393. As it is verified by the Metamath program, isosctrlem1ALT 41261 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 41261. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isosctrlem1ALT | ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10589 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
3 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | subcld 10991 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ) |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ) |
6 | subeq0 10906 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
7 | 6 | biimpd 231 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
8 | 7 | idiALT 40804 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
9 | 1, 3, 8 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
10 | 9 | con3d 155 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → ¬ (1 − 𝐴) = 0)) |
11 | df-ne 3017 | . . . . . . . 8 ⊢ ((1 − 𝐴) ≠ 0 ↔ ¬ (1 − 𝐴) = 0) | |
12 | 11 | biimpri 230 | . . . . . . 7 ⊢ (¬ (1 − 𝐴) = 0 → (1 − 𝐴) ≠ 0) |
13 | 10, 12 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → (1 − 𝐴) ≠ 0)) |
14 | 13 | imp 409 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0) |
15 | 5, 14 | logcld 25148 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ) |
16 | 15 | imcld 14548 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
17 | 16 | 3adant2 1127 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
18 | pire 25038 | . . . . 5 ⊢ π ∈ ℝ | |
19 | 2re 11705 | . . . . 5 ⊢ 2 ∈ ℝ | |
20 | 2ne0 11735 | . . . . 5 ⊢ 2 ≠ 0 | |
21 | 18, 19, 20 | redivcli 11401 | . . . 4 ⊢ (π / 2) ∈ ℝ |
22 | 21 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (π / 2) ∈ ℝ) |
23 | 18 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → π ∈ ℝ) |
24 | neghalfpirx 25046 | . . . 4 ⊢ -(π / 2) ∈ ℝ* | |
25 | 21 | rexri 10693 | . . . 4 ⊢ (π / 2) ∈ ℝ* |
26 | 3 | recld 14547 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
27 | 26 | recnd 10663 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
28 | 27 | subidd 10979 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
29 | 28 | adantr 483 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
30 | 1re 10635 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . 9 ⊢ (1 ∈ ℂ → 1 ∈ ℝ) |
32 | 1, 31 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ∈ ℝ |
33 | 3 | releabsd 14805 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
34 | 33 | adantr 483 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
35 | id 22 | . . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → (abs‘𝐴) = 1) | |
36 | 35 | adantl 484 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1) |
37 | 34, 36 | breqtrd 5084 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1) |
38 | lesub1 11128 | . . . . . . . . . 10 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ≤ 1 ↔ ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))) | |
39 | 38 | 3impcombi 41144 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
40 | 39 | idiALT 40804 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
41 | 32, 26, 37, 40 | mp3an2ani 1464 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
42 | 29, 41 | eqbrtrrd 5082 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴))) |
43 | 32 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → 1 ∈ ℝ) |
44 | 43 | rered 14577 | . . . . . . . . . 10 ⊢ (⊤ → (ℜ‘1) = 1) |
45 | 44 | mptru 1540 | . . . . . . . . 9 ⊢ (ℜ‘1) = 1 |
46 | oveq1 7157 | . . . . . . . . . 10 ⊢ ((ℜ‘1) = 1 → ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴))) | |
47 | 46 | eqcomd 2827 | . . . . . . . . 9 ⊢ ((ℜ‘1) = 1 → (1 − (ℜ‘𝐴)) = ((ℜ‘1) − (ℜ‘𝐴))) |
48 | 45, 47 | ax-mp 5 | . . . . . . . 8 ⊢ (1 − (ℜ‘𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)) |
49 | resub 14480 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴))) | |
50 | 49 | eqcomd 2827 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
51 | 50 | idiALT 40804 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
52 | 1, 3, 51 | sylancr 589 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
53 | 48, 52 | syl5eq 2868 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
54 | 53 | adantr 483 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
55 | 42, 54 | breqtrd 5084 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴))) |
56 | argrege0 25188 | . . . . . . 7 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) | |
57 | 56 | 3coml 1123 | . . . . . 6 ⊢ (((1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ∈ ℂ) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) |
58 | 57 | 3com13 1120 | . . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ≠ 0) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) |
59 | 4, 55, 14, 58 | eel12131 41040 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) |
60 | iccleub 12786 | . . . 4 ⊢ ((-(π / 2) ∈ ℝ* ∧ (π / 2) ∈ ℝ* ∧ (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) | |
61 | 24, 25, 59, 60 | mp3an12i 1461 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) |
62 | pipos 25040 | . . . . . 6 ⊢ 0 < π | |
63 | 18, 62 | elrpii 12386 | . . . . 5 ⊢ π ∈ ℝ+ |
64 | rphalflt 12412 | . . . . 5 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
65 | 63, 64 | ax-mp 5 | . . . 4 ⊢ (π / 2) < π |
66 | 65 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (π / 2) < π) |
67 | 17, 22, 23, 61, 66 | lelttrd 10792 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π) |
68 | 17, 67 | ltned 10770 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 − cmin 10864 -cneg 10865 / cdiv 11291 2c2 11686 ℝ+crp 12383 [,]cicc 12735 ℜcre 14450 ℑcim 14451 abscabs 14587 πcpi 15414 logclog 25132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 |
This theorem is referenced by: (None) |
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