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Theorem lognegb 22817
Description: If a number has imaginary part equal to  pi, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)
Assertion
Ref Expression
lognegb  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )

Proof of Theorem lognegb
StepHypRef Expression
1 logneg 22815 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
21fveq2d 5875 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  ( Im `  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) ) )
3 relogcl 22806 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u A )  e.  RR )
4 pire 22695 . . . . 5  |-  pi  e.  RR
5 crim 12923 . . . . 5  |-  ( ( ( log `  -u A
)  e.  RR  /\  pi  e.  RR )  -> 
( Im `  (
( log `  -u A
)  +  ( _i  x.  pi ) ) )  =  pi )
63, 4, 5sylancl 662 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( ( log `  -u A )  +  ( _i  x.  pi ) ) )  =  pi )
72, 6eqtrd 2508 . . 3  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  pi )
8 negneg 9879 . . . . . . 7  |-  ( A  e.  CC  ->  -u -u A  =  A )
98adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u A  =  A
)
109fveq2d 5875 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  -u -u A
)  =  ( log `  A ) )
1110fveq2d 5875 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  -u -u A ) )  =  ( Im `  ( log `  A ) ) )
1211eqeq1d 2469 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  -u -u A
) )  =  pi  <->  ( Im `  ( log `  A ) )  =  pi ) )
137, 12syl5ib 219 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  ->  ( Im `  ( log `  A ) )  =  pi ) )
14 logcl 22799 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
1514replimd 13005 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  =  ( ( Re `  ( log `  A ) )  +  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )
1615fveq2d 5875 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
17 eflog 22807 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
1814recld 13002 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  RR )
1918recnd 9632 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  CC )
20 ax-icn 9561 . . . . . . 7  |-  _i  e.  CC
2114imcld 13003 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  RR )
2221recnd 9632 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  CC )
23 mulcl 9586 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  ( log `  A ) )  e.  CC )  ->  (
_i  x.  ( Im `  ( log `  A
) ) )  e.  CC )
2420, 22, 23sylancr 663 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  (
Im `  ( log `  A ) ) )  e.  CC )
25 efadd 13703 . . . . . 6  |-  ( ( ( Re `  ( log `  A ) )  e.  CC  /\  (
_i  x.  ( Im `  ( log `  A
) ) )  e.  CC )  ->  ( exp `  ( ( Re
`  ( log `  A
) )  +  ( _i  x.  ( Im
`  ( log `  A
) ) ) ) )  =  ( ( exp `  ( Re
`  ( log `  A
) ) )  x.  ( exp `  (
_i  x.  ( Im `  ( log `  A
) ) ) ) ) )
2619, 24, 25syl2anc 661 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  ( exp `  (
_i  x.  ( Im `  ( log `  A
) ) ) ) ) )
2716, 17, 263eqtr3d 2516 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
28 oveq2 6302 . . . . . . . 8  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( _i  x.  ( Im `  ( log `  A ) ) )  =  ( _i  x.  pi ) )
2928fveq2d 5875 . . . . . . 7  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  ( exp `  ( _i  x.  pi ) ) )
30 efipi 22709 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
3129, 30syl6eq 2524 . . . . . 6  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  -u 1
)
3231oveq2d 6310 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) )
3332eqeq2d 2481 . . . 4  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )  <->  A  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) ) )
3427, 33syl5ibcom 220 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  A ) )  =  pi  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) ) )
3518rpefcld 13713 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  RR+ )
3635rpcnd 11268 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  CC )
37 neg1cn 10649 . . . . . . . . 9  |-  -u 1  e.  CC
38 mulcom 9588 . . . . . . . . 9  |-  ( ( ( exp `  (
Re `  ( log `  A ) ) )  e.  CC  /\  -u 1  e.  CC )  ->  (
( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( -u 1  x.  ( exp `  (
Re `  ( log `  A ) ) ) ) )
3936, 37, 38sylancl 662 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( -u 1  x.  ( exp `  (
Re `  ( log `  A ) ) ) ) )
4036mulm1d 10018 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u 1  x.  ( exp `  ( Re `  ( log `  A ) ) ) )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4139, 40eqtrd 2508 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4241negeqd 9824 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u -u ( exp `  (
Re `  ( log `  A ) ) ) )
4336negnegd 9931 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u ( exp `  (
Re `  ( log `  A ) ) )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4442, 43eqtrd 2508 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4544, 35eqeltrd 2555 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  e.  RR+ )
46 negeq 9822 . . . . 5  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  -u A  = 
-u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) )
4746eleq1d 2536 . . . 4  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  ( -u A  e.  RR+  <->  -u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  e.  RR+ )
)
4845, 47syl5ibrcom 222 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  ->  -u A  e.  RR+ )
)
4934, 48syld 44 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  A ) )  =  pi  ->  -u A  e.  RR+ ) )
5013, 49impbid 191 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593  (class class class)co 6294   CCcc 9500   RRcr 9501   0cc0 9502   1c1 9503   _ici 9504    + caddc 9505    x. cmul 9507   -ucneg 9816   RR+crp 11230   Recre 12905   Imcim 12906   expce 13671   picpi 13676   logclog 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581  ax-mulf 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-fi 7881  df-sup 7911  df-oi 7945  df-card 8330  df-cda 8558  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  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 11803  df-fl 11907  df-mod 11975  df-seq 12086  df-exp 12145  df-fac 12332  df-bc 12359  df-hash 12384  df-shft 12875  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-limsup 13269  df-clim 13286  df-rlim 13287  df-sum 13484  df-ef 13677  df-sin 13679  df-cos 13680  df-pi 13682  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-hom 14591  df-cco 14592  df-rest 14690  df-topn 14691  df-0g 14709  df-gsum 14710  df-topgen 14711  df-pt 14712  df-prds 14715  df-xrs 14769  df-qtop 14774  df-imas 14775  df-xps 14777  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-mulg 15909  df-cntz 16204  df-cmn 16650  df-psmet 18258  df-xmet 18259  df-met 18260  df-bl 18261  df-mopn 18262  df-fbas 18263  df-fg 18264  df-cnfld 18268  df-top 19245  df-bases 19247  df-topon 19248  df-topsp 19249  df-cld 19365  df-ntr 19366  df-cls 19367  df-nei 19444  df-lp 19482  df-perf 19483  df-cn 19573  df-cnp 19574  df-haus 19661  df-tx 19908  df-hmeo 20101  df-fil 20192  df-fm 20284  df-flim 20285  df-flf 20286  df-xms 20668  df-ms 20669  df-tms 20670  df-cncf 21227  df-limc 22115  df-dv 22116  df-log 22787
This theorem is referenced by:  logcj  22834  argimgt0  22840  dvloglem  22872  logf1o2  22874  ang180lem2  22985  logrec  22994  angpieqvdlem2  23003  asinneg  23060
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