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Theorem eflogeq 22852
Description: Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
eflogeq  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( exp `  A
)  =  B  <->  E. n  e.  ZZ  A  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
Distinct variable groups:    A, n    B, n

Proof of Theorem eflogeq
StepHypRef Expression
1 efcl 13697 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efne0 13710 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
31, 2logcld 22824 . . . . . . . 8  |-  ( A  e.  CC  ->  ( log `  ( exp `  A
) )  e.  CC )
4 efsub 13713 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
53, 4mpdan 668 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
6 eflog 22830 . . . . . . . . 9  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A )  =/=  0 )  -> 
( exp `  ( log `  ( exp `  A
) ) )  =  ( exp `  A
) )
71, 2, 6syl2anc 661 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( log `  ( exp `  A ) ) )  =  ( exp `  A ) )
87oveq2d 6311 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A
)  /  ( exp `  A ) ) )
91, 2dividd 10330 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  A ) )  =  1 )
105, 8, 93eqtrd 2512 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1 )
11 subcl 9831 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
123, 11mpdan 668 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
13 efeq1 22782 . . . . . . 7  |-  ( ( A  -  ( log `  ( exp `  A
) ) )  e.  CC  ->  ( ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1  <->  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
1412, 13syl 16 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( A  -  ( log `  ( exp `  A
) ) ) )  =  1  <->  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
1510, 14mpbid 210 . . . . 5  |-  ( A  e.  CC  ->  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ )
16 ax-icn 9563 . . . . . . . . . 10  |-  _i  e.  CC
17 2cn 10618 . . . . . . . . . . 11  |-  2  e.  CC
18 picn 22719 . . . . . . . . . . 11  |-  pi  e.  CC
1917, 18mulcli 9613 . . . . . . . . . 10  |-  ( 2  x.  pi )  e.  CC
2016, 19mulcli 9613 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
2120a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
22 ine0 10004 . . . . . . . . . 10  |-  _i  =/=  0
23 2ne0 10640 . . . . . . . . . . 11  |-  2  =/=  0
24 pire 22718 . . . . . . . . . . . 12  |-  pi  e.  RR
25 pipos 22720 . . . . . . . . . . . 12  |-  0  <  pi
2624, 25gt0ne0ii 10101 . . . . . . . . . . 11  |-  pi  =/=  0
2717, 18, 23, 26mulne0i 10204 . . . . . . . . . 10  |-  ( 2  x.  pi )  =/=  0
2816, 19, 22, 27mulne0i 10204 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
2928a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
3012, 21, 29divcan2d 10334 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( A  -  ( log `  ( exp `  A
) ) ) )
3130oveq2d 6311 . . . . . 6  |-  ( A  e.  CC  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) ) ) )  =  ( ( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A
) ) ) ) )
32 pncan3 9840 . . . . . . 7  |-  ( ( ( log `  ( exp `  A ) )  e.  CC  /\  A  e.  CC )  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
333, 32mpancom 669 . . . . . 6  |-  ( A  e.  CC  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
3431, 33eqtr2d 2509 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( log `  ( exp `  A
) )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) ) ) )
35 oveq2 6303 . . . . . . . 8  |-  ( n  =  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n )  =  ( ( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) ) )
3635oveq2d 6311 . . . . . . 7  |-  ( n  =  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) )  ->  ( ( log `  ( exp `  A
) )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) ) ) )
3736eqeq2d 2481 . . . . . 6  |-  ( n  =  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) )  ->  ( A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  <->  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) ) ) ) ) )
3837rspcev 3219 . . . . 5  |-  ( ( ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) )  e.  ZZ  /\  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A ) ) )  /  ( _i  x.  ( 2  x.  pi ) ) ) ) ) )  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
3915, 34, 38syl2anc 661 . . . 4  |-  ( A  e.  CC  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
40393ad2ant1 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
41 fveq2 5872 . . . . . 6  |-  ( ( exp `  A )  =  B  ->  ( log `  ( exp `  A
) )  =  ( log `  B ) )
4241oveq1d 6310 . . . . 5  |-  ( ( exp `  A )  =  B  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
4342eqeq2d 2481 . . . 4  |-  ( ( exp `  A )  =  B  ->  ( A  =  ( ( log `  ( exp `  A
) )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  <->  A  =  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
4443rexbidv 2978 . . 3  |-  ( ( exp `  A )  =  B  ->  ( E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A
) )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  <->  E. n  e.  ZZ  A  =  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
) ) ) )
4540, 44syl5ibcom 220 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( exp `  A
)  =  B  ->  E. n  e.  ZZ  A  =  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
) ) ) )
46 logcl 22822 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
47463adant1 1014 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( log `  B )  e.  CC )
4847adantr 465 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( log `  B
)  e.  CC )
49 zcn 10881 . . . . . . . 8  |-  ( n  e.  ZZ  ->  n  e.  CC )
5049adantl 466 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  n  e.  CC )
51 mulcl 9588 . . . . . . 7  |-  ( ( ( _i  x.  (
2  x.  pi ) )  e.  CC  /\  n  e.  CC )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
5220, 50, 51sylancr 663 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
53 efadd 13708 . . . . . 6  |-  ( ( ( log `  B
)  e.  CC  /\  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )  ->  ( exp `  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )  =  ( ( exp `  ( log `  B ) )  x.  ( exp `  (
( _i  x.  (
2  x.  pi ) )  x.  n ) ) ) )
5448, 52, 53syl2anc 661 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( exp `  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )  =  ( ( exp `  ( log `  B ) )  x.  ( exp `  (
( _i  x.  (
2  x.  pi ) )  x.  n ) ) ) )
55 eflog 22830 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( exp `  ( log `  B ) )  =  B )
56553adant1 1014 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( exp `  ( log `  B
) )  =  B )
57 ef2kpi 22737 . . . . . 6  |-  ( n  e.  ZZ  ->  ( exp `  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  1 )
5856, 57oveqan12d 6314 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( ( exp `  ( log `  B ) )  x.  ( exp `  (
( _i  x.  (
2  x.  pi ) )  x.  n ) ) )  =  ( B  x.  1 ) )
59 simpl2 1000 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  B  e.  CC )
6059mulid1d 9625 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( B  x.  1 )  =  B )
6154, 58, 603eqtrd 2512 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( exp `  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )  =  B )
62 fveq2 5872 . . . . 5  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( exp `  A )  =  ( exp `  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
6362eqeq1d 2469 . . . 4  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( ( exp `  A )  =  B  <->  ( exp `  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )  =  B ) )
6461, 63syl5ibrcom 222 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( A  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  ->  ( exp `  A
)  =  B ) )
6564rexlimdva 2959 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( E. n  e.  ZZ  A  =  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
) )  ->  ( exp `  A )  =  B ) )
6645, 65impbid 191 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( exp `  A
)  =  B  <->  E. n  e.  ZZ  A  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505   _ici 9506    + caddc 9507    x. cmul 9509    - cmin 9817    / cdiv 10218   2c2 10597   ZZcz 10876   expce 13676   picpi 13681   logclog 22808
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684  df-cos 13685  df-pi 13687  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139  df-log 22810
This theorem is referenced by:  cxpeq  22997
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