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Theorem lognegb 19937
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 19935 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u -u A )  =  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) )
21fveq2d 5489 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  ( Im `  ( ( log `  -u A
)  +  ( _i  x.  pi ) ) ) )
3 relogcl 19926 . . . . 5  |-  ( -u A  e.  RR+  ->  ( log `  -u A )  e.  RR )
4 pire 19826 . . . . 5  |-  pi  e.  RR
5 crim 11594 . . . . 5  |-  ( ( ( log `  -u A
)  e.  RR  /\  pi  e.  RR )  -> 
( Im `  (
( log `  -u A
)  +  ( _i  x.  pi ) ) )  =  pi )
63, 4, 5sylancl 645 . . . 4  |-  ( -u A  e.  RR+  ->  (
Im `  ( ( log `  -u A )  +  ( _i  x.  pi ) ) )  =  pi )
72, 6eqtrd 2316 . . 3  |-  ( -u A  e.  RR+  ->  (
Im `  ( log `  -u -u A ) )  =  pi )
8 negneg 9092 . . . . . . 7  |-  ( A  e.  CC  ->  -u -u A  =  A )
98adantr 453 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u A  =  A
)
109fveq2d 5489 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  -u -u A
)  =  ( log `  A ) )
1110fveq2d 5489 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  -u -u A ) )  =  ( Im `  ( log `  A ) ) )
1211eqeq1d 2292 . . 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 19920 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  e.  CC )
1514replimd 11676 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( log `  A
)  =  ( ( Re `  ( log `  A ) )  +  ( _i  x.  (
Im `  ( log `  A ) ) ) ) )
1615fveq2d 5489 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  ( exp `  (
( Re `  ( log `  A ) )  +  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
17 eflog 19927 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  ( log `  A ) )  =  A )
1814recld 11673 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  RR )
1918recnd 8856 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Re `  ( log `  A ) )  e.  CC )
20 ax-icn 8791 . . . . . . 7  |-  _i  e.  CC
2114imcld 11674 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  RR )
2221recnd 8856 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( Im `  ( log `  A ) )  e.  CC )
23 mulcl 8816 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( Im `  ( log `  A ) )  e.  CC )  ->  (
_i  x.  ( Im `  ( log `  A
) ) )  e.  CC )
2420, 22, 23sylancr 646 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( _i  x.  (
Im `  ( log `  A ) ) )  e.  CC )
25 efadd 12369 . . . . . 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 644 . . . . 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 2324 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) ) )
28 oveq2 5827 . . . . . . . 8  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( _i  x.  ( Im `  ( log `  A ) ) )  =  ( _i  x.  pi ) )
2928fveq2d 5489 . . . . . . 7  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  ( exp `  ( _i  x.  pi ) ) )
30 efipi 19835 . . . . . . 7  |-  ( exp `  ( _i  x.  pi ) )  =  -u
1
3129, 30syl6eq 2332 . . . . . 6  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( exp `  ( _i  x.  (
Im `  ( log `  A ) ) ) )  =  -u 1
)
3231oveq2d 5835 . . . . 5  |-  ( ( Im `  ( log `  A ) )  =  pi  ->  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  ( exp `  ( _i  x.  ( Im `  ( log `  A ) ) ) ) )  =  ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 ) )
3332eqeq2d 2295 . . . 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 12379 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  RR+ )
3635rpcnd 10387 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( exp `  (
Re `  ( log `  A ) ) )  e.  CC )
37 neg1cn 9808 . . . . . . . . 9  |-  -u 1  e.  CC
38 mulcom 8818 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( -u 1  x.  ( exp `  (
Re `  ( log `  A ) ) ) ) )
4036mulm1d 9226 . . . . . . . 8  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( -u 1  x.  ( exp `  ( Re `  ( log `  A ) ) ) )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4139, 40eqtrd 2316 . . . . . . 7  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u ( exp `  (
Re `  ( log `  A ) ) ) )
4241negeqd 9041 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  = 
-u -u ( exp `  (
Re `  ( log `  A ) ) ) )
4336negnegd 9143 . . . . . 6  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u -u ( exp `  (
Re `  ( log `  A ) ) )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4442, 43eqtrd 2316 . . . . 5  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  =  ( exp `  (
Re `  ( log `  A ) ) ) )
4544, 35eqeltrd 2358 . . . 4  |-  ( ( A  e.  CC  /\  A  =/=  0 )  ->  -u ( ( exp `  (
Re `  ( log `  A ) ) )  x.  -u 1 )  e.  RR+ )
46 negeq 9039 . . . . 5  |-  ( A  =  ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
)  ->  -u A  = 
-u ( ( exp `  ( Re `  ( log `  A ) ) )  x.  -u 1
) )
4746eleq1d 2350 . . . 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 1624    e. wcel 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5819   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733   _ici 8734    + caddc 8735    x. cmul 8737   -ucneg 9033   RR+crp 10349   Recre 11576   Imcim 11577   expce 12337   picpi 12342   logclog 19906
This theorem is referenced by:  logcj  19954  argimgt0  19960  dvloglem  19989  logf1o2  19991  ang180lem2  20102  logrec  20111  angpieqvdlem2  20120  asinneg  20176
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-sum 12153  df-ef 12343  df-sin 12345  df-cos 12346  df-pi 12348  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908
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