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Theorem lognegb 22043
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 22041 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
21fveq2d 5700 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  ( Im `  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) ) )
3 relogcl 22032 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u A )  e.  RR )
4 pire 21926 . . . . 5  |-  pi  e.  RR
5 crim 12609 . . . . 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 2475 . . 3  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  pi )
8 negneg 9664 . . . . . . 7  |-  ( A  e.  CC  ->  -u -u A  =  A )
98adantr 465 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u A  =  A
)
109fveq2d 5700 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  -u -u A
)  =  ( log `  A ) )
1110fveq2d 5700 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  -u -u A ) )  =  ( Im `  ( log `  A ) ) )
1211eqeq1d 2451 . . 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 22025 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
1514replimd 12691 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  =  ( ( Re `  ( log `  A ) )  +  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )
1615fveq2d 5700 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
17 eflog 22033 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
1814recld 12688 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  RR )
1918recnd 9417 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  CC )
20 ax-icn 9346 . . . . . . 7  |-  _i  e.  CC
2114imcld 12689 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  RR )
2221recnd 9417 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  CC )
23 mulcl 9371 . . . . . . 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 13384 . . . . . 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 2483 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
28 oveq2 6104 . . . . . . . 8  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( _i  x.  ( Im `  ( log `  A ) ) )  =  ( _i  x.  pi ) )
2928fveq2d 5700 . . . . . . 7  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  ( exp `  ( _i  x.  pi ) ) )
30 efipi 21940 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
3129, 30syl6eq 2491 . . . . . 6  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  -u 1
)
3231oveq2d 6112 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) )
3332eqeq2d 2454 . . . 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 13394 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  RR+ )
3635rpcnd 11034 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  CC )
37 neg1cn 10430 . . . . . . . . 9  |-  -u 1  e.  CC
38 mulcom 9373 . . . . . . . . 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 9801 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u 1  x.  ( exp `  ( Re `  ( log `  A ) ) ) )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4139, 40eqtrd 2475 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4241negeqd 9609 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u -u ( exp `  (
Re `  ( log `  A ) ) ) )
4336negnegd 9715 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u ( exp `  (
Re `  ( log `  A ) ) )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4442, 43eqtrd 2475 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4544, 35eqeltrd 2517 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  e.  RR+ )
46 negeq 9607 . . . . 5  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  -u A  = 
-u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) )
4746eleq1d 2509 . . . 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 1369    e. wcel 1756    =/= wne 2611   ` cfv 5423  (class class class)co 6096   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288   _ici 9289    + caddc 9290    x. cmul 9292   -ucneg 9601   RR+crp 10996   Recre 12591   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:  logcj  22060  argimgt0  22066  dvloglem  22098  logf1o2  22100  ang180lem2  22211  logrec  22220  angpieqvdlem2  22229  asinneg  22286
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