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Theorem logrec 22174
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 9997 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  CC )
2 recne0 10003 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =/=  0 )
3 eflog 21987 . . . . . . . 8  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
41, 2, 3syl2anc 656 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  ( 1  /  A ) ) )  =  ( 1  /  A ) )
54eqcomd 2446 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  A
)  =  ( exp `  ( log `  (
1  /  A ) ) ) )
65oveq2d 6106 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  ( 1  /  ( exp `  ( log `  ( 1  /  A ) ) ) ) )
7 eflog 21987 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
8 recrec 10024 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( 1  /  (
1  /  A ) )  =  A )
97, 8eqtr4d 2476 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( 1  / 
( 1  /  A
) ) )
101, 2logcld 21981 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  CC )
11 efneg 13378 . . . . . 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 2483 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
14133adant3 1003 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( exp `  ( log `  A
) )  =  ( exp `  -u ( log `  ( 1  /  A ) ) ) )
1514fveq2d 5692 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  ( exp `  ( log `  A ) ) )  =  ( log `  ( exp `  -u ( log `  ( 1  /  A ) ) ) ) )
16 logrncl 21978 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  ran  log )
17163adant3 1003 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  ( log `  A )  e. 
ran  log )
18 logef 21989 . . 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 2606 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =/= 
pi 
<->  -.  ( Im `  ( log `  A ) )  =  pi )
21 lognegb 21997 . . . . . . . . . . . 12  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( -u (
1  /  A )  e.  RR+  <->  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi ) )
221, 2, 21syl2anc 656 . . . . . . . . . . 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 9336 . . . . . . . . . . . 12  |-  1  e.  CC
25 divneg2 10051 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  -u (
1  /  A )  =  ( 1  /  -u A ) )
2624, 25mp3an1 1296 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( 1  /  A
)  =  ( 1  /  -u A ) )
2726eleq1d 2507 . . . . . . . . . 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 9606 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  -u A  e.  CC )
3029adantr 462 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  e.  CC )
31 negeq0 9659 . . . . . . . . . . . 12  |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
3231necon3bid 2641 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  =/=  0  <->  -u A  =/=  0 ) )
3332biimpa 481 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u A  =/=  0 )
34 rpreccl 11010 . . . . . . . . . . 11  |-  ( ( 1  /  -u A
)  e.  RR+  ->  ( 1  /  ( 1  /  -u A ) )  e.  RR+ )
35 recrec 10024 . . . . . . . . . . . 12  |-  ( (
-u A  e.  CC  /\  -u A  =/=  0
)  ->  ( 1  /  ( 1  /  -u A ) )  = 
-u A )
3635eleq1d 2507 . . . . . . . . . . 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 656 . . . . . . . . 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 21997 . . . . . . . 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 1179 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  A ) )  =  pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
4420, 43syl3an3b 1251 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )
45 logrncl 21978 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  CC  /\  ( 1  /  A
)  =/=  0 )  ->  ( log `  (
1  /  A ) )  e.  ran  log )
461, 2, 45syl2anc 656 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  (
1  /  A ) )  e.  ran  log )
47 logreclem 22173 . . . . . 6  |-  ( ( ( log `  (
1  /  A ) )  e.  ran  log  /\ 
-.  ( Im `  ( log `  ( 1  /  A ) ) )  =  pi )  ->  -u ( log `  (
1  /  A ) )  e.  ran  log )
4846, 47sylan 468 . . . . 5  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  -.  ( Im
`  ( log `  (
1  /  A ) ) )  =  pi )  ->  -u ( log `  ( 1  /  A
) )  e.  ran  log )
49483impa 1177 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  -.  ( Im `  ( log `  ( 1  /  A
) ) )  =  pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
5044, 49syld3an3 1258 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  (
Im `  ( log `  A ) )  =/= 
pi )  ->  -u ( log `  ( 1  /  A ) )  e. 
ran  log )
51 logef 21989 . . 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 2481 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   ran crn 4837   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   -ucneg 9592    / cdiv 9989   RR+crp 10987   Imcim 12583   expce 13343   picpi 13348   logclog 21965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967
This theorem is referenced by:  isosctrlem2  22176  logbrec  26400
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