isosctrlem1ALT - Mathbox for Alan Sare

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Theorem isosctrlem1ALT 45384

Description: Lemma for isosctr 26810. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 26810. As it is verified by the Metamath program, isosctrlem1ALT 45384 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 45384. (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 11094	. . . . . . . 8 ⊢ 1 ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
4	2, 3	subcld 11503	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ)
5	4	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
6		subeq0 11418	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
7	6	biimpd 230	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴))
8	7	idiALT 44929	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 → 1 = 𝐴))
9	1, 3, 8	sylancr 593	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((1 − 𝐴) = 0 → 1 = 𝐴))
10	9	con3d 152	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → ¬ (1 − 𝐴) = 0))
11		df-ne 2936	. . . . . . . 8 ⊢ ((1 − 𝐴) ≠ 0 ↔ ¬ (1 − 𝐴) = 0)
12	11	biimpri 229	. . . . . . 7 ⊢ (¬ (1 − 𝐴) = 0 → (1 − 𝐴) ≠ 0)
13	10, 12	syl6 35	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (¬ 1 = 𝐴 → (1 − 𝐴) ≠ 0))
14	13	imp 407	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
15	5, 14	logcld 26559	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
16	15	imcld 15155	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
17	16	3adant2 1137	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
18		pire 26446	. . . . 5 ⊢ π ∈ ℝ
19		2re 12253	. . . . 5 ⊢ 2 ∈ ℝ
20		2ne0 12283	. . . . 5 ⊢ 2 ≠ 0
21	18, 19, 20	redivcli 11920	. . . 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 26455	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
25	21	rexri 11201	. . . 4 ⊢ (π / 2) ∈ ℝ^*
26	3	recld 15154	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
27	26	recnd 11171	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ)
28	27	subidd 11491	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
29	28	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
30		1re 11142	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ (1 ∈ ℂ → 1 ∈ ℝ)
32	1, 31	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℝ
33	3	releabsd 15414	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴))
34	33	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴))
35		id 22	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → (abs‘𝐴) = 1)
36	35	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (abs‘𝐴) = 1)
37	34, 36	breqtrd 5105	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1)
38		lesub1 11642	. . . . . . . . . 10 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ≤ 1 ↔ ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))))
39	38	3impcombi 45267	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
40	39	idiALT 44929	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
41	32, 26, 37, 40	mp3an2ani 1476	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
42	29, 41	eqbrtrrd 5103	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴)))
43	32	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 ∈ ℝ)
44	43	rered 15184	. . . . . . . . . 10 ⊢ (⊤ → (ℜ‘1) = 1)
45	44	mptru 1554	. . . . . . . . 9 ⊢ (ℜ‘1) = 1
46		oveq1 7370	. . . . . . . . . 10 ⊢ ((ℜ‘1) = 1 → ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴)))
47	46	eqcomd 2746	. . . . . . . . 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 15087	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)))
50	49	eqcomd 2746	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
51	50	idiALT 44929	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
52	1, 3, 51	sylancr 593	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (1 − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
55	42, 54	breqtrd 5105	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴)))
56		argrege0 26600	. . . . . . 7 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
57	56	3coml 1133	. . . . . 6 ⊢ (((1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ∈ ℂ) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
58	57	3com13 1130	. . . . 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 45163	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
60		iccleub 13352	. . . 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 1473	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
62		pipos 26448	. . . . . 6 ⊢ 0 < π
63	18, 62	elrpii 12943	. . . . 5 ⊢ π ∈ ℝ⁺
64		rphalflt 12971	. . . . 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 11302	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π)
68	17, 67	ltned 11280	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 396 ∧ w3a 1092 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ≠ wne 2935 class class class wbr 5079 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 -cneg 11376 / cdiv 11805 2c2 12234 ℝ⁺crp 12940 [,]cicc 13299 ℜcre 15057 ℑcim 15058 abscabs 15194 πcpi 16029 logclog 26543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-ef 16030 df-sin 16032 df-cos 16033 df-pi 16035 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-log 26545

This theorem is referenced by: (None)

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