Theorem logrec 24501

Description: Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)

Assertion

Ref	Expression
logrec	|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) )

Proof of Theorem logrec

Step	Hyp	Ref	Expression
1		reccl 10692	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC )
2		recne0 10698	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) =/= 0 )
3		eflog 24323	. . . . . . . 8 |- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( exp ` ( log ` ( 1 / A ) ) ) = ( 1 / A ) )
4	1, 2, 3	syl2anc 693	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` ( 1 / A ) ) ) = ( 1 / A ) )
5	4	eqcomd 2628	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) = ( exp ` ( log ` ( 1 / A ) ) ) )
6	5	oveq2d 6666	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / ( 1 / A ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) )
7		eflog 24323	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A )
8		recrec 10722	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / ( 1 / A ) ) = A )
9	7, 8	eqtr4d 2659	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = ( 1 / ( 1 / A ) ) )
10	1, 2	logcld 24317	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( log ` ( 1 / A ) ) e. CC )
11		efneg 14828	. . . . . 6 |- ( ( log ` ( 1 / A ) ) e. CC -> ( exp ` -u ( log ` ( 1 / A ) ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) )
12	10, 11	syl 17	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` -u ( log ` ( 1 / A ) ) ) = ( 1 / ( exp ` ( log ` ( 1 / A ) ) ) ) )
13	6, 9, 12	3eqtr4d 2666	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = ( exp ` -u ( log ` ( 1 / A ) ) ) )
14	13	3adant3 1081	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( exp ` ( log ` A ) ) = ( exp ` -u ( log ` ( 1 / A ) ) ) )
15	14	fveq2d 6195	. 2 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) )
16		logrncl 24314	. . . 4 |- ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. ran log )
17	16	3adant3 1081	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` A ) e. ran log )
18		logef 24328	. . 3 |- ( ( log ` A ) e. ran log -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` A ) )
19	17, 18	syl 17	. 2 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` ( exp ` ( log ` A ) ) ) = ( log ` A ) )
20		df-ne 2795	. . . . 5 |- ( ( Im ` ( log ` A ) ) =/= pi <-> -. ( Im ` ( log ` A ) ) = pi )
21		lognegb 24336	. . . . . . . . . . . 12 |- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( Im ` ( log ` ( 1 / A ) ) ) = pi ) )
22	1, 2, 21	syl2anc 693	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( Im ` ( log ` ( 1 / A ) ) ) = pi ) )
23	22	biimprd 238	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = pi -> -u ( 1 / A ) e. RR+ ) )
24		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
25		divneg2 10749	. . . . . . . . . . . 12 |- ( ( 1 e. CC /\ A e. CC /\ A =/= 0 ) -> -u ( 1 / A ) = ( 1 / -u A ) )
26	24, 25	mp3an1 1411	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> -u ( 1 / A ) = ( 1 / -u A ) )
27	26	eleq1d 2686	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> ( -u ( 1 / A ) e. RR+ <-> ( 1 / -u A ) e. RR+ ) )
28	23, 27	sylibd 229	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = pi -> ( 1 / -u A ) e. RR+ ) )
29		negcl 10281	. . . . . . . . . . 11 |- ( A e. CC -> -u A e. CC )
30	29	adantr 481	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> -u A e. CC )
31		negeq0 10335	. . . . . . . . . . . 12 |- ( A e. CC -> ( A = 0 <-> -u A = 0 ) )
32	31	necon3bid 2838	. . . . . . . . . . 11 |- ( A e. CC -> ( A =/= 0 <-> -u A =/= 0 ) )
33	32	biimpa 501	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> -u A =/= 0 )
34		rpreccl 11857	. . . . . . . . . . 11 |- ( ( 1 / -u A ) e. RR+ -> ( 1 / ( 1 / -u A ) ) e. RR+ )
35		recrec 10722	. . . . . . . . . . . 12 |- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( 1 / ( 1 / -u A ) ) = -u A )
36	35	eleq1d 2686	. . . . . . . . . . 11 |- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( ( 1 / ( 1 / -u A ) ) e. RR+ <-> -u A e. RR+ ) )
37	34, 36	syl5ib 234	. . . . . . . . . 10 |- ( ( -u A e. CC /\ -u A =/= 0 ) -> ( ( 1 / -u A ) e. RR+ -> -u A e. RR+ ) )
38	30, 33, 37	syl2anc 693	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( 1 / -u A ) e. RR+ -> -u A e. RR+ ) )
39	28, 38	syld 47	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = pi -> -u A e. RR+ ) )
40		lognegb 24336	. . . . . . . 8 |- ( ( A e. CC /\ A =/= 0 ) -> ( -u A e. RR+ <-> ( Im ` ( log ` A ) ) = pi ) )
41	39, 40	sylibd 229	. . . . . . 7 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( Im ` ( log ` ( 1 / A ) ) ) = pi -> ( Im ` ( log ` A ) ) = pi ) )
42	41	con3d 148	. . . . . 6 |- ( ( A e. CC /\ A =/= 0 ) -> ( -. ( Im ` ( log ` A ) ) = pi -> -. ( Im ` ( log ` ( 1 / A ) ) ) = pi ) )
43	42	3impia 1261	. . . . 5 |- ( ( A e. CC /\ A =/= 0 /\ -. ( Im ` ( log ` A ) ) = pi ) -> -. ( Im ` ( log ` ( 1 / A ) ) ) = pi )
44	20, 43	syl3an3b 1364	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> -. ( Im ` ( log ` ( 1 / A ) ) ) = pi )
45		logrncl 24314	. . . . . 6 |- ( ( ( 1 / A ) e. CC /\ ( 1 / A ) =/= 0 ) -> ( log ` ( 1 / A ) ) e. ran log )
46	1, 2, 45	syl2anc 693	. . . . 5 |- ( ( A e. CC /\ A =/= 0 ) -> ( log ` ( 1 / A ) ) e. ran log )
47		logreclem 24500	. . . . 5 |- ( ( ( log ` ( 1 / A ) ) e. ran log /\ -. ( Im ` ( log ` ( 1 / A ) ) ) = pi ) -> -u ( log ` ( 1 / A ) ) e. ran log )
48	46, 47	stoic3 1701	. . . 4 |- ( ( A e. CC /\ A =/= 0 /\ -. ( Im ` ( log ` ( 1 / A ) ) ) = pi ) -> -u ( log ` ( 1 / A ) ) e. ran log )
49	44, 48	syld3an3 1371	. . 3 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> -u ( log ` ( 1 / A ) ) e. ran log )
50		logef 24328	. . 3 |- ( -u ( log ` ( 1 / A ) ) e. ran log -> ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) = -u ( log ` ( 1 / A ) ) )
51	49, 50	syl 17	. 2 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` ( exp ` -u ( log ` ( 1 / A ) ) ) ) = -u ( log ` ( 1 / A ) ) )
52	15, 19, 51	3eqtr3d 2664	1 |- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ran crn 5115   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   -ucneg 10267   / cdiv 10684   RR+crp 11832   Imcim 13838   expce 14792   picpi 14797   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  logbrec  24520  isosctrlem2  24549