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Theorem lognegb 19891
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 19889 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
21fveq2d 5448 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  ( Im `  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) ) )
3 relogcl 19880 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u A )  e.  RR )
4 pire 19780 . . . . 5  |-  pi  e.  RR
5 crim 11551 . . . . 5  |-  ( ( ( log `  -u A
)  e.  RR  /\  pi  e.  RR )  -> 
( Im `  (
( log `  -u A
)  +  ( _i  x.  pi ) ) )  =  pi )
63, 4, 5sylancl 646 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( ( log `  -u A )  +  ( _i  x.  pi ) ) )  =  pi )
72, 6eqtrd 2288 . . 3  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  pi )
8 negneg 9051 . . . . . . 7  |-  ( A  e.  CC  ->  -u -u A  =  A )
98adantr 453 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u A  =  A
)
109fveq2d 5448 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  -u -u A
)  =  ( log `  A ) )
1110fveq2d 5448 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  -u -u A ) )  =  ( Im `  ( log `  A ) ) )
1211eqeq1d 2264 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  -u -u A
) )  =  pi  <->  ( Im `  ( log `  A ) )  =  pi ) )
137, 12syl5ib 212 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  ->  ( Im `  ( log `  A ) )  =  pi ) )
14 logcl 19874 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
1514replimd 11633 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  =  ( ( Re `  ( log `  A ) )  +  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )
1615fveq2d 5448 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
17 eflog 19881 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
1814recld 11630 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  RR )
1918recnd 8815 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  CC )
20 ax-icn 8750 . . . . . . 7  |-  _i  e.  CC
2114imcld 11631 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  RR )
2221recnd 8815 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  CC )
23 mulcl 8775 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  ( log `  A ) )  e.  CC )  ->  (
_i  x.  ( Im `  ( log `  A
) ) )  e.  CC )
2420, 22, 23sylancr 647 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  (
Im `  ( log `  A ) ) )  e.  CC )
25 efadd 12323 . . . . . 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 645 . . . . 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 2296 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
28 oveq2 5786 . . . . . . . 8  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( _i  x.  ( Im `  ( log `  A ) ) )  =  ( _i  x.  pi ) )
2928fveq2d 5448 . . . . . . 7  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  ( exp `  ( _i  x.  pi ) ) )
30 efipi 19789 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
3129, 30syl6eq 2304 . . . . . 6  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  -u 1
)
3231oveq2d 5794 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) )
3332eqeq2d 2267 . . . 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 213 . . 3  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( Im `  ( log `  A ) )  =  pi  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) ) )
3518rpefcld 12333 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  RR+ )
3635rpcnd 10345 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  CC )
37 neg1cn 9767 . . . . . . . . 9  |-  -u 1  e.  CC
38 mulcom 8777 . . . . . . . . 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 646 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( -u 1  x.  ( exp `  (
Re `  ( log `  A ) ) ) ) )
4036mulm1d 9185 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u 1  x.  ( exp `  ( Re `  ( log `  A ) ) ) )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4139, 40eqtrd 2288 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4241negeqd 9000 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u -u ( exp `  (
Re `  ( log `  A ) ) ) )
4336negnegd 9102 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u ( exp `  (
Re `  ( log `  A ) ) )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4442, 43eqtrd 2288 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4544, 35eqeltrd 2330 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  e.  RR+ )
46 negeq 8998 . . . . 5  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  -u A  = 
-u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) )
4746eleq1d 2322 . . . 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 215 . . 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 185 1  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u A  e.  RR+  <->  (
Im `  ( log `  A ) )  =  pi ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692   _ici 8693    + caddc 8694    x. cmul 8696   -ucneg 8992   RR+crp 10307   Recre 11533   Imcim 11534   expce 12291   picpi 12296   logclog 19860
This theorem is referenced by:  logcj  19908  argimgt0  19914  dvloglem  19943  logf1o2  19945  ang180lem2  20056  logrec  20065  angpieqvdlem2  20074  asinneg  20130
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862
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