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Theorem logrec 22220
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 10006 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
2 recne0 10012 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
3 eflog 22033 . . . . . . . 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 2448 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =  ( exp `  ( log `  (
1  /  A ) ) ) )
65oveq2d 6112 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  ( 1  /  ( exp `  ( log `  ( 1  /  A ) ) ) ) )
7 eflog 22033 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
8 recrec 10033 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
97, 8eqtr4d 2478 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( 1  / 
( 1  /  A
) ) )
101, 2logcld 22027 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  CC )
11 efneg 13387 . . . . . 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 2485 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
14133adant3 1008 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( exp `  ( log `  A
) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
1514fveq2d 5700 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) ) )
16 logrncl 22024 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  ran  log )
17163adant3 1008 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  e. 
ran  log )
18 logef 22035 . . 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 2613 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =/= 
pi 
<->  -.  ( Im `  ( log `  A ) )  =  pi )
21 lognegb 22043 . . . . . . . . . . . 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 9345 . . . . . . . . . . . 12  |-  1  e.  CC
25 divneg2 10060 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
2624, 25mp3an1 1301 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( 1  /  A
)  =  ( 1  /  -u A ) )
2726eleq1d 2509 . . . . . . . . . 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 9615 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
3029adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  e.  CC )
31 negeq0 9668 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3231necon3bid 2648 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3332biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
34 rpreccl 11019 . . . . . . . . . . 11  |-  ( ( 1  /  -u A
)  e.  RR+  ->  ( 1  /  ( 1  /  -u A ) )  e.  RR+ )
35 recrec 10033 . . . . . . . . . . . 12  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( 1  /  ( 1  /  -u A ) )  = 
-u A )
3635eleq1d 2509 . . . . . . . . . . 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 22043 . . . . . . . 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 1184 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  A ) )  =  pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
4420, 43syl3an3b 1256 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
45 logrncl 22024 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  ran  log )
461, 2, 45syl2anc 661 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  ran  log )
47 logreclem 22219 . . . . . 6  |-  ( ( ( log `  (
1  /  A ) )  e.  ran  log  /\ 
-.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi )  ->  -u ( log `  (
1  /  A ) )  e.  ran  log )
4846, 47sylan 471 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  -.  ( Im
`  ( log `  (
1  /  A ) ) )  =  pi )  ->  -u ( log `  ( 1  /  A
) )  e.  ran  log )
49483impa 1182 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
5044, 49syld3an3 1263 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
51 logef 22035 . . 3  |-  ( -u ( log `  ( 1  /  A ) )  e.  ran  log  ->  ( log `  ( exp `  -u ( log `  (
1  /  A ) ) ) )  = 
-u ( log `  (
1  /  A ) ) )
5250, 51syl 16 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) )  =  -u ( log `  ( 1  /  A ) ) )
5315, 19, 523eqtr3d 2483 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   ran crn 4846   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288   -ucneg 9601    / cdiv 9998   RR+crp 10996   Imcim 12592   expce 13352   picpi 13357   logclog 22011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013
This theorem is referenced by:  isosctrlem2  22222  logbrec  26469
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