isosctrlem1ALT - Mathbox for Alan Sare

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Theorem isosctrlem1ALT 44959

Description: Lemma for isosctr 26865. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 26865. As it is verified by the Metamath program, isosctrlem1ALT 44959 verifies https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 44959. (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 11214	. . . . . . . 8 ⊢ 1 ∈ ℂ
2	1	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ)
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
4	2, 3	subcld 11621	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ)
5	4	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
6		subeq0 11536	. . . . . . . . . . 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 44503	. . . . . . . . 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 2940	. . . . . . . 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 26613	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
16	15	imcld 15235	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
17	16	3adant2 1131	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
18		pire 26501	. . . . 5 ⊢ π ∈ ℝ
19		2re 12341	. . . . 5 ⊢ 2 ∈ ℝ
20		2ne0 12371	. . . . 5 ⊢ 2 ≠ 0
21	18, 19, 20	redivcli 12035	. . . 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 26509	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
25	21	rexri 11320	. . . 4 ⊢ (π / 2) ∈ ℝ^*
26	3	recld 15234	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
27	26	recnd 11290	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ)
28	27	subidd 11609	. . . . . . . 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 11262	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
31	30	a1i 11	. . . . . . . . 9 ⊢ (1 ∈ ℂ → 1 ∈ ℝ)
32	1, 31	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℝ
33	3	releabsd 15491	. . . . . . . . . 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 5168	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1)
38		lesub1 11758	. . . . . . . . . 10 ⊢ (((ℜ‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ) → ((ℜ‘𝐴) ≤ 1 ↔ ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))))
39	38	3impcombi 44842	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
40	39	idiALT 44503	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ (ℜ‘𝐴) ∈ ℝ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
41	32, 26, 37, 40	mp3an2ani 1469	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
42	29, 41	eqbrtrrd 5166	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴)))
43	32	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → 1 ∈ ℝ)
44	43	rered 15264	. . . . . . . . . 10 ⊢ (⊤ → (ℜ‘1) = 1)
45	44	mptru 1546	. . . . . . . . 9 ⊢ (ℜ‘1) = 1
46		oveq1 7439	. . . . . . . . . 10 ⊢ ((ℜ‘1) = 1 → ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴)))
47	46	eqcomd 2742	. . . . . . . . 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 15167	. . . . . . . . . . 11 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)))
50	49	eqcomd 2742	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((ℜ‘1) − (ℜ‘𝐴)) = (ℜ‘(1 − 𝐴)))
51	50	idiALT 44503	. . . . . . . . 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 2788	. . . . . . 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 5168	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴)))
56		argrege0 26654	. . . . . . 7 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
57	56	3coml 1127	. . . . . 6 ⊢ (((1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴)) ∧ (1 − 𝐴) ∈ ℂ) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
58	57	3com13 1124	. . . . 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 44738	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
60		iccleub 13443	. . . 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 1466	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
62		pipos 26503	. . . . . 6 ⊢ 0 < π
63	18, 62	elrpii 13038	. . . . 5 ⊢ π ∈ ℝ⁺
64		rphalflt 13065	. . . . 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 11420	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π)
68	17, 67	ltned 11398	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 1539 ⊤wtru 1540 ∈ wcel 2107 ≠ wne 2939 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 − cmin 11493 -cneg 11494 / cdiv 11921 2c2 12322 ℝ⁺crp 13035 [,]cicc 13391 ℜcre 15137 ℑcim 15138 abscabs 15274 πcpi 16103 logclog 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 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 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599

This theorem is referenced by: (None)

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