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Theorem lognegb 20352
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 20350 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
21fveq2d 5673 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  ( Im `  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) ) )
3 relogcl 20341 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u A )  e.  RR )
4 pire 20240 . . . . 5  |-  pi  e.  RR
5 crim 11848 . . . . 5  |-  ( ( ( log `  -u A
)  e.  RR  /\  pi  e.  RR )  -> 
( Im `  (
( log `  -u A
)  +  ( _i  x.  pi ) ) )  =  pi )
63, 4, 5sylancl 644 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( ( log `  -u A )  +  ( _i  x.  pi ) ) )  =  pi )
72, 6eqtrd 2420 . . 3  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  pi )
8 negneg 9284 . . . . . . 7  |-  ( A  e.  CC  ->  -u -u A  =  A )
98adantr 452 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u A  =  A
)
109fveq2d 5673 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  -u -u A
)  =  ( log `  A ) )
1110fveq2d 5673 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  -u -u A ) )  =  ( Im `  ( log `  A ) ) )
1211eqeq1d 2396 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  -u -u A
) )  =  pi  <->  ( Im `  ( log `  A ) )  =  pi ) )
137, 12syl5ib 211 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  ->  ( Im `  ( log `  A ) )  =  pi ) )
14 logcl 20334 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
1514replimd 11930 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  =  ( ( Re `  ( log `  A ) )  +  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )
1615fveq2d 5673 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
17 eflog 20342 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
1814recld 11927 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  RR )
1918recnd 9048 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  CC )
20 ax-icn 8983 . . . . . . 7  |-  _i  e.  CC
2114imcld 11928 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  RR )
2221recnd 9048 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  CC )
23 mulcl 9008 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  ( log `  A ) )  e.  CC )  ->  (
_i  x.  ( Im `  ( log `  A
) ) )  e.  CC )
2420, 22, 23sylancr 645 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  (
Im `  ( log `  A ) ) )  e.  CC )
25 efadd 12624 . . . . . 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 643 . . . . 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 2428 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
28 oveq2 6029 . . . . . . . 8  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( _i  x.  ( Im `  ( log `  A ) ) )  =  ( _i  x.  pi ) )
2928fveq2d 5673 . . . . . . 7  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  ( exp `  ( _i  x.  pi ) ) )
30 efipi 20249 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
3129, 30syl6eq 2436 . . . . . 6  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  -u 1
)
3231oveq2d 6037 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) )
3332eqeq2d 2399 . . . 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 212 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  A ) )  =  pi  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) ) )
3518rpefcld 12634 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  RR+ )
3635rpcnd 10583 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  CC )
37 neg1cn 10000 . . . . . . . . 9  |-  -u 1  e.  CC
38 mulcom 9010 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( -u 1  x.  ( exp `  (
Re `  ( log `  A ) ) ) ) )
4036mulm1d 9418 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u 1  x.  ( exp `  ( Re `  ( log `  A ) ) ) )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4139, 40eqtrd 2420 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4241negeqd 9233 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u -u ( exp `  (
Re `  ( log `  A ) ) ) )
4336negnegd 9335 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u ( exp `  (
Re `  ( log `  A ) ) )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4442, 43eqtrd 2420 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4544, 35eqeltrd 2462 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  e.  RR+ )
46 negeq 9231 . . . . 5  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  -u A  = 
-u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) )
4746eleq1d 2454 . . . 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 214 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( A  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  ->  -u A  e.  RR+ )
)
4934, 48syld 42 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  A ) )  =  pi  ->  -u A  e.  RR+ ) )
5013, 49impbid 184 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925   _ici 8926    + caddc 8927    x. cmul 8929   -ucneg 9225   RR+crp 10545   Recre 11830   Imcim 11831   expce 12592   picpi 12597   logclog 20320
This theorem is referenced by:  logcj  20369  argimgt0  20375  dvloglem  20407  logf1o2  20409  ang180lem2  20520  logrec  20529  angpieqvdlem2  20538  asinneg  20594
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322
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