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Theorem logrec 23127
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
StepHypRef Expression
1 reccl 10221 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
2 recne0 10227 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
3 eflog 22940 . . . . . . . 8  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
41, 2, 3syl2anc 661 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
54eqcomd 2451 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =  ( exp `  ( log `  (
1  /  A ) ) ) )
65oveq2d 6297 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  ( 1  /  ( exp `  ( log `  ( 1  /  A ) ) ) ) )
7 eflog 22940 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
8 recrec 10248 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
97, 8eqtr4d 2487 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( 1  / 
( 1  /  A
) ) )
101, 2logcld 22934 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  CC )
11 efneg 13814 . . . . . 6  |-  ( ( log `  ( 1  /  A ) )  e.  CC  ->  ( exp `  -u ( log `  (
1  /  A ) ) )  =  ( 1  /  ( exp `  ( log `  (
1  /  A ) ) ) ) )
1210, 11syl 16 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  -u ( log `  ( 1  /  A ) ) )  =  ( 1  / 
( exp `  ( log `  ( 1  /  A ) ) ) ) )
136, 9, 123eqtr4d 2494 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
14133adant3 1017 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( exp `  ( log `  A
) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
1514fveq2d 5860 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) ) )
16 logrncl 22931 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  ran  log )
17163adant3 1017 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  e. 
ran  log )
18 logef 22942 . . 3  |-  ( ( log `  A )  e.  ran  log  ->  ( log `  ( exp `  ( log `  A
) ) )  =  ( log `  A
) )
1917, 18syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  A ) )
20 df-ne 2640 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =/= 
pi 
<->  -.  ( Im `  ( log `  A ) )  =  pi )
21 lognegb 22950 . . . . . . . . . . . 12  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( -u (
1  /  A )  e.  RR+  <->  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
221, 2, 21syl2anc 661 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
Im `  ( log `  ( 1  /  A
) ) )  =  pi ) )
2322biimprd 223 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u ( 1  /  A
)  e.  RR+ )
)
24 ax-1cn 9553 . . . . . . . . . . . 12  |-  1  e.  CC
25 divneg2 10275 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
2624, 25mp3an1 1312 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( 1  /  A
)  =  ( 1  /  -u A ) )
2726eleq1d 2512 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u ( 1  /  A )  e.  RR+  <->  (
1  /  -u A
)  e.  RR+ )
)
2823, 27sylibd 214 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( 1  /  -u A
)  e.  RR+ )
)
29 negcl 9825 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
3029adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  e.  CC )
31 negeq0 9878 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3231necon3bid 2701 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3332biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
34 rpreccl 11253 . . . . . . . . . . 11  |-  ( ( 1  /  -u A
)  e.  RR+  ->  ( 1  /  ( 1  /  -u A ) )  e.  RR+ )
35 recrec 10248 . . . . . . . . . . . 12  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( 1  /  ( 1  /  -u A ) )  = 
-u A )
3635eleq1d 2512 . . . . . . . . . . 11  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  ( 1  /  -u A ) )  e.  RR+  <->  -u A  e.  RR+ ) )
3734, 36syl5ib 219 . . . . . . . . . 10  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( (
1  /  -u A
)  e.  RR+  ->  -u A  e.  RR+ ) )
3830, 33, 37syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( 1  /  -u A )  e.  RR+  -> 
-u A  e.  RR+ ) )
3928, 38syld 44 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  -u A  e.  RR+ ) )
40 lognegb 22950 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
4139, 40sylibd 214 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  ( 1  /  A ) ) )  =  pi  ->  ( Im `  ( log `  A ) )  =  pi ) )
4241con3d 133 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -.  ( Im
`  ( log `  A
) )  =  pi 
->  -.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
43423impia 1194 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  A ) )  =  pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
4420, 43syl3an3b 1267 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
45 logrncl 22931 . . . . . 6  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  ran  log )
461, 2, 45syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  ran  log )
47 logreclem 23126 . . . . 5  |-  ( ( ( log `  (
1  /  A ) )  e.  ran  log  /\ 
-.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi )  ->  -u ( log `  (
1  /  A ) )  e.  ran  log )
4846, 47stoic3 1596 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
4944, 48syld3an3 1274 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
50 logef 22942 . . 3  |-  ( -u ( log `  ( 1  /  A ) )  e.  ran  log  ->  ( log `  ( exp `  -u ( log `  (
1  /  A ) ) ) )  = 
-u ( log `  (
1  /  A ) ) )
5149, 50syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) )  =  -u ( log `  ( 1  /  A ) ) )
5215, 19, 513eqtr3d 2492 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  = 
-u ( log `  (
1  /  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   ran crn 4990   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   -ucneg 9811    / cdiv 10213   RR+crp 11230   Imcim 12912   expce 13778   picpi 13783   logclog 22918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10986  df-uz 11092  df-q 11193  df-rp 11231  df-xneg 11328  df-xadd 11329  df-xmul 11330  df-ioo 11543  df-ioc 11544  df-ico 11545  df-icc 11546  df-fz 11683  df-fzo 11806  df-fl 11910  df-mod 11978  df-seq 12089  df-exp 12148  df-fac 12335  df-bc 12362  df-hash 12387  df-shft 12881  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-limsup 13275  df-clim 13292  df-rlim 13293  df-sum 13490  df-ef 13784  df-sin 13786  df-cos 13787  df-pi 13789  df-struct 14615  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-mulr 14692  df-starv 14693  df-sca 14694  df-vsca 14695  df-ip 14696  df-tset 14697  df-ple 14698  df-ds 14700  df-unif 14701  df-hom 14702  df-cco 14703  df-rest 14801  df-topn 14802  df-0g 14820  df-gsum 14821  df-topgen 14822  df-pt 14823  df-prds 14826  df-xrs 14880  df-qtop 14885  df-imas 14886  df-xps 14888  df-mre 14964  df-mrc 14965  df-acs 14967  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-submnd 15945  df-mulg 16038  df-cntz 16333  df-cmn 16778  df-psmet 18389  df-xmet 18390  df-met 18391  df-bl 18392  df-mopn 18393  df-fbas 18394  df-fg 18395  df-cnfld 18399  df-top 19376  df-bases 19378  df-topon 19379  df-topsp 19380  df-cld 19497  df-ntr 19498  df-cls 19499  df-nei 19576  df-lp 19614  df-perf 19615  df-cn 19705  df-cnp 19706  df-haus 19793  df-tx 20040  df-hmeo 20233  df-fil 20324  df-fm 20416  df-flim 20417  df-flf 20418  df-xms 20800  df-ms 20801  df-tms 20802  df-cncf 21359  df-limc 22247  df-dv 22248  df-log 22920
This theorem is referenced by:  isosctrlem2  23129  logbrec  27998
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