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Theorem logreclem 26267

Description: Symmetry of the natural logarithm range by negation. Lemma for logrec 26268. (Contributed by Saveliy Skresanov, 27-Dec-2016.)

**Assertion**
Ref	Expression
logreclem	⊢ ((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)

**Proof of Theorem logreclem**
Step	Hyp	Ref	Expression
1		logrncn 26071	. . . . . . . . 9 ⊢ (𝐴 ∈ ran log → 𝐴 ∈ ℂ)
2	1	adantr 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → 𝐴 ∈ ℂ)
3	2	negcld 11558	. . . . . . 7 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → -𝐴 ∈ ℂ)
4		ellogrn 26068	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))
5	4	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ran log → (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))
6	5	simp3d 1145	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ran log → (ℑ‘𝐴) ≤ π)
7		imcl 15058	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) ∈ ℝ)
8		pire 25968	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
9		leneg 11717	. . . . . . . . . . . . . 14 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝐴) ≤ π ↔ -π ≤ -(ℑ‘𝐴)))
10	9	biimpd 228	. . . . . . . . . . . . 13 ⊢ (((ℑ‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → ((ℑ‘𝐴) ≤ π → -π ≤ -(ℑ‘𝐴)))
11	7, 8, 10	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → ((ℑ‘𝐴) ≤ π → -π ≤ -(ℑ‘𝐴)))
12	1, 6, 11	sylc 65	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ran log → -π ≤ -(ℑ‘𝐴))
13	8	renegcli 11521	. . . . . . . . . . . . . 14 ⊢ -π ∈ ℝ
14	13	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → -π ∈ ℝ)
15	7	renegcld 11641	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → -(ℑ‘𝐴) ∈ ℝ)
16	14, 15	leloed 11357	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (-π ≤ -(ℑ‘𝐴) ↔ (-π < -(ℑ‘𝐴) ∨ -π = -(ℑ‘𝐴))))
17	16	biimpd 228	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (-π ≤ -(ℑ‘𝐴) → (-π < -(ℑ‘𝐴) ∨ -π = -(ℑ‘𝐴))))
18	1, 12, 17	sylc 65	. . . . . . . . . 10 ⊢ (𝐴 ∈ ran log → (-π < -(ℑ‘𝐴) ∨ -π = -(ℑ‘𝐴)))
19	18	orcomd 870	. . . . . . . . 9 ⊢ (𝐴 ∈ ran log → (-π = -(ℑ‘𝐴) ∨ -π < -(ℑ‘𝐴)))
20	19	orcanai 1002	. . . . . . . 8 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → -π < -(ℑ‘𝐴))
21	5	simp2d 1144	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ran log → -π < (ℑ‘𝐴))
22		ltnegcon1 11715	. . . . . . . . . . . . 13 ⊢ ((π ∈ ℝ ∧ (ℑ‘𝐴) ∈ ℝ) → (-π < (ℑ‘𝐴) ↔ -(ℑ‘𝐴) < π))
23	22	biimpd 228	. . . . . . . . . . . 12 ⊢ ((π ∈ ℝ ∧ (ℑ‘𝐴) ∈ ℝ) → (-π < (ℑ‘𝐴) → -(ℑ‘𝐴) < π))
24	8, 7, 23	sylancr 588	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (-π < (ℑ‘𝐴) → -(ℑ‘𝐴) < π))
25	1, 21, 24	sylc 65	. . . . . . . . . 10 ⊢ (𝐴 ∈ ran log → -(ℑ‘𝐴) < π)
26	25	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → -(ℑ‘𝐴) < π)
27		ltle 11302	. . . . . . . . . . . 12 ⊢ ((-(ℑ‘𝐴) ∈ ℝ ∧ π ∈ ℝ) → (-(ℑ‘𝐴) < π → -(ℑ‘𝐴) ≤ π))
28	15, 8, 27	sylancl 587	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (-(ℑ‘𝐴) < π → -(ℑ‘𝐴) ≤ π))
29	1, 28	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ran log → (-(ℑ‘𝐴) < π → -(ℑ‘𝐴) ≤ π))
30	29	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → (-(ℑ‘𝐴) < π → -(ℑ‘𝐴) ≤ π))
31	26, 30	mpd 15	. . . . . . . 8 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → -(ℑ‘𝐴) ≤ π)
32		imneg 15080	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℑ‘-𝐴) = -(ℑ‘𝐴))
33	32	breq2d 5161	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-π < (ℑ‘-𝐴) ↔ -π < -(ℑ‘𝐴)))
34	2, 33	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → (-π < (ℑ‘-𝐴) ↔ -π < -(ℑ‘𝐴)))
35	32	breq1d 5159	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((ℑ‘-𝐴) ≤ π ↔ -(ℑ‘𝐴) ≤ π))
36	2, 35	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → ((ℑ‘-𝐴) ≤ π ↔ -(ℑ‘𝐴) ≤ π))
37	34, 36	anbi12d 632	. . . . . . . 8 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → ((-π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π) ↔ (-π < -(ℑ‘𝐴) ∧ -(ℑ‘𝐴) ≤ π)))
38	20, 31, 37	mpbir2and 712	. . . . . . 7 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → (-π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π))
39		3anass 1096	. . . . . . 7 ⊢ ((-𝐴 ∈ ℂ ∧ -π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π) ↔ (-𝐴 ∈ ℂ ∧ (-π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π)))
40	3, 38, 39	sylanbrc 584	. . . . . 6 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → (-𝐴 ∈ ℂ ∧ -π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π))
41		ellogrn 26068	. . . . . 6 ⊢ (-𝐴 ∈ ran log ↔ (-𝐴 ∈ ℂ ∧ -π < (ℑ‘-𝐴) ∧ (ℑ‘-𝐴) ≤ π))
42	40, 41	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ ran log ∧ ¬ -π = -(ℑ‘𝐴)) → -𝐴 ∈ ran log)
43	42	ex 414	. . . 4 ⊢ (𝐴 ∈ ran log → (¬ -π = -(ℑ‘𝐴) → -𝐴 ∈ ran log))
44	43	orrd 862	. . 3 ⊢ (𝐴 ∈ ran log → (-π = -(ℑ‘𝐴) ∨ -𝐴 ∈ ran log))
45		recn 11200	. . . . . . . 8 ⊢ (π ∈ ℝ → π ∈ ℂ)
46		recn 11200	. . . . . . . 8 ⊢ ((ℑ‘𝐴) ∈ ℝ → (ℑ‘𝐴) ∈ ℂ)
47	45, 46	anim12i 614	. . . . . . 7 ⊢ ((π ∈ ℝ ∧ (ℑ‘𝐴) ∈ ℝ) → (π ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ))
48	8, 7, 47	sylancr 588	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (π ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ))
49	1, 48	syl 17	. . . . 5 ⊢ (𝐴 ∈ ran log → (π ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ))
50		neg11 11511	. . . . . 6 ⊢ ((π ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (-π = -(ℑ‘𝐴) ↔ π = (ℑ‘𝐴)))
51		eqcom 2740	. . . . . 6 ⊢ (π = (ℑ‘𝐴) ↔ (ℑ‘𝐴) = π)
52	50, 51	bitrdi 287	. . . . 5 ⊢ ((π ∈ ℂ ∧ (ℑ‘𝐴) ∈ ℂ) → (-π = -(ℑ‘𝐴) ↔ (ℑ‘𝐴) = π))
53	49, 52	syl 17	. . . 4 ⊢ (𝐴 ∈ ran log → (-π = -(ℑ‘𝐴) ↔ (ℑ‘𝐴) = π))
54	53	orbi1d 916	. . 3 ⊢ (𝐴 ∈ ran log → ((-π = -(ℑ‘𝐴) ∨ -𝐴 ∈ ran log) ↔ ((ℑ‘𝐴) = π ∨ -𝐴 ∈ ran log)))
55	44, 54	mpbid 231	. 2 ⊢ (𝐴 ∈ ran log → ((ℑ‘𝐴) = π ∨ -𝐴 ∈ ran log))
56	55	orcanai 1002	1 ⊢ ((𝐴 ∈ ran log ∧ ¬ (ℑ‘𝐴) = π) → -𝐴 ∈ ran log)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ran crn 5678 ‘cfv 6544 ℂcc 11108 ℝcr 11109 < clt 11248 ≤ cle 11249 -cneg 11445 ℑcim 15045 πcpi 16010 logclog 26063

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065

This theorem is referenced by: logrec 26268

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