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Mirrors > Home > MPE Home > Th. List > Mathboxes > isosctrlem1ALT | Structured version Visualization version GIF version |
Description: Lemma for isosctr 26879. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 26879. As it is verified by the Metamath program, isosctrlem1ALT 44932 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 44932. (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 11211 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
3 | id 22 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
4 | 2, 3 | subcld 11618 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ) |
5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ) |
6 | subeq0 11533 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
7 | 6 | biimpd 229 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
8 | 7 | idiALT 44475 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
9 | 1, 3, 8 | sylancr 587 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴)) |
10 | 9 | con3d 152 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → ¬ (1 − 𝐴) = 0)) |
11 | df-ne 2939 | . . . . . . . 8 ⊢ ((1 − 𝐴) ≠ 0 ↔ ¬ (1 − 𝐴) = 0) | |
12 | 11 | biimpri 228 | . . . . . . 7 ⊢ (¬ (1 − 𝐴) = 0 → (1 − 𝐴) ≠ 0) |
13 | 10, 12 | syl6 35 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → (1 − 𝐴) ≠ 0)) |
14 | 13 | imp 406 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0) |
15 | 5, 14 | logcld 26627 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ) |
16 | 15 | imcld 15231 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
17 | 16 | 3adant2 1130 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
18 | pire 26515 | . . . . 5 ⊢ π ∈ ℝ | |
19 | 2re 12338 | . . . . 5 ⊢ 2 ∈ ℝ | |
20 | 2ne0 12368 | . . . . 5 ⊢ 2 ≠ 0 | |
21 | 18, 19, 20 | redivcli 12032 | . . . 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 26523 | . . . 4 ⊢ -(π / 2) ∈ ℝ* | |
25 | 21 | rexri 11317 | . . . 4 ⊢ (π / 2) ∈ ℝ* |
26 | 3 | recld 15230 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) |
27 | 26 | recnd 11287 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
28 | 27 | subidd 11606 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
29 | 28 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
30 | 1re 11259 | . . . . . . . . . 10 ⊢ 1 ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . . . . 9 ⊢ (1 ∈ ℂ → 1 ∈ ℝ) |
32 | 1, 31 | ax-mp 5 | . . . . . . . 8 ⊢ 1 ∈ ℝ |
33 | 3 | releabsd 15487 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
34 | 33 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
35 | id 22 | . . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → (abs‘𝐴) = 1) | |
36 | 35 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1) |
37 | 34, 36 | breqtrd 5174 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1) |
38 | lesub1 11755 | . . . . . . . . . 10 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ≤ 1 ↔ ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))) | |
39 | 38 | 3impcombi 44815 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
40 | 39 | idiALT 44475 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
41 | 32, 26, 37, 40 | mp3an2ani 1467 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
42 | 29, 41 | eqbrtrrd 5172 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴))) |
43 | 32 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → 1 ∈ ℝ) |
44 | 43 | rered 15260 | . . . . . . . . . 10 ⊢ (⊤ → (ℜ‘1) = 1) |
45 | 44 | mptru 1544 | . . . . . . . . 9 ⊢ (ℜ‘1) = 1 |
46 | oveq1 7438 | . . . . . . . . . 10 ⊢ ((ℜ‘1) = 1 → ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴))) | |
47 | 46 | eqcomd 2741 | . . . . . . . . 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 15163 | . . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴))) | |
50 | 49 | eqcomd 2741 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
51 | 50 | idiALT 44475 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
52 | 1, 3, 51 | sylancr 587 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
53 | 48, 52 | eqtrid 2787 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (1 − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
54 | 53 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴))) |
55 | 42, 54 | breqtrd 5174 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴))) |
56 | argrege0 26668 | . . . . . . 7 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) | |
57 | 56 | 3coml 1126 | . . . . . 6 ⊢ (((1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ∈ ℂ) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) |
58 | 57 | 3com13 1123 | . . . . 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 44711 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) |
60 | iccleub 13439 | . . . 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 1464 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) |
62 | pipos 26517 | . . . . . 6 ⊢ 0 < π | |
63 | 18, 62 | elrpii 13035 | . . . . 5 ⊢ π ∈ ℝ+ |
64 | rphalflt 13062 | . . . . 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 11417 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π) |
68 | 17, 67 | ltned 11395 | 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 395 ∧ w3a 1086 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 2c2 12319 ℝ+crp 13032 [,]cicc 13387 ℜcre 15133 ℑcim 15134 abscabs 15270 πcpi 16099 logclog 26611 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 |
This theorem is referenced by: (None) |
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