isosctrlem1ALT - Mathbox for Alan Sare

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Theorem isosctrlem1ALT 39484

Description: Lemma for isosctr 24596. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart http://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html. As it is verified by the Metamath program, isosctrlem1ALT 39484 verifies http://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

**Assertion**
Ref	Expression
isosctrlem1ALT	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

**Proof of Theorem isosctrlem1ALT**
Step	Hyp	Ref	Expression
1		ax-1cn 10032	. . . . . . . 8 ⊢ 1 ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
4	2, 3	subcld 10430	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ)
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
6		subeq0 10345	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
7	6	biimpd 219	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴))
8	7	idiALT 39000	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴))
9	1, 3, 8	sylancr 696	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
10	9	con3d 148	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → ¬ (1 − 𝐴) = 0))
11		df-ne 2824	. . . . . . . 8 ⊢ ((1 − 𝐴) ≠ 0 ↔ ¬ (1 − 𝐴) = 0)
12	11	biimpri 218	. . . . . . 7 ⊢ (¬ (1 − 𝐴) = 0 → (1 − 𝐴) ≠ 0)
13	10, 12	syl6 35	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → (1 − 𝐴) ≠ 0))
14	13	imp 444	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
15	5, 14	logcld 24362	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
16	15	imcld 13979	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
17	16	3adant2 1100	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
18		pire 24255	. . . . 5 ⊢ π ∈ ℝ
19		2re 11128	. . . . 5 ⊢ 2 ∈ ℝ
20		2ne0 11151	. . . . 5 ⊢ 2 ≠ 0
21	18, 19, 20	redivcli 10830	. . . 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 24263	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
25	21	rexri 10135	. . . 4 ⊢ (π / 2) ∈ ℝ^*
26	3	recld 13978	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
27	26	recnd 10106	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ)
28	27	subidd 10418	. . . . . . . 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 10077	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ (1 ∈ ℂ → 1 ∈ ℝ)
32	1, 31	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℝ
33	3	releabsd 14234	. . . . . . . . . 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 4711	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1)
38		lesub1 10560	. . . . . . . . . 10 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ≤ 1 ↔ ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))))
39	38	3impcombi 39361	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
40	39	idiALT 39000	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
41	32, 26, 37, 40	mp3an2ani 1471	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
42	29, 41	eqbrtrrd 4709	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴)))
43	32	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 ∈ ℝ)
44	43	rered 14008	. . . . . . . . . 10 ⊢ (⊤ → (ℜ‘1) = 1)
45	44	trud 1533	. . . . . . . . 9 ⊢ (ℜ‘1) = 1
46		oveq1 6697	. . . . . . . . . 10 ⊢ ((ℜ‘1) = 1 → ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴)))
47	46	eqcomd 2657	. . . . . . . . 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 13911	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)))
50	49	eqcomd 2657	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
51	50	idiALT 39000	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
52	1, 3, 51	sylancr 696	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
53	48, 52	syl5eq 2697	. . . . . . 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 4711	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴)))
56		argrege0 24402	. . . . . . 7 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
57	56	3coml 1292	. . . . . 6 ⊢ (((1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ∈ ℂ) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
58	57	3com13 1289	. . . . 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 39255	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
60		iccleub 12267	. . . 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 1468	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
62		pipos 24257	. . . . . 6 ⊢ 0 < π
63	18, 62	elrpii 11873	. . . . 5 ⊢ π ∈ ℝ⁺
64		rphalflt 11898	. . . . 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 10233	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π)
68	17, 67	ltned 10211	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 383 ∧ w3a 1054 = wceq 1523 ⊤wtru 1524 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 ℝ^*cxr 10111 < clt 10112 ≤ cle 10113 − cmin 10304 -cneg 10305 / cdiv 10722 2c2 11108 ℝ⁺crp 11870 [,]cicc 12216 ℜcre 13881 ℑcim 13882 abscabs 14018 πcpi 14841 logclog 24346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-mulg 17588 df-cntz 17796 df-cmn 18241 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-xms 22172 df-ms 22173 df-tms 22174 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348

This theorem is referenced by: (None)

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