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Theorem eflogeq 22964
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 13800 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efne0 13814 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
31, 2logcld 22936 . . . . . . . 8  |-  ( A  e.  CC  ->  ( log `  ( exp `  A
) )  e.  CC )
4 efsub 13817 . . . . . . . 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 22942 . . . . . . . . 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 6297 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A
)  /  ( exp `  A ) ) )
91, 2dividd 10325 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  A ) )  =  1 )
105, 8, 93eqtrd 2488 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1 )
11 subcl 9824 . . . . . . . 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 22894 . . . . . . 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 9554 . . . . . . . . . 10  |-  _i  e.  CC
17 2cn 10613 . . . . . . . . . . 11  |-  2  e.  CC
18 picn 22830 . . . . . . . . . . 11  |-  pi  e.  CC
1917, 18mulcli 9604 . . . . . . . . . 10  |-  ( 2  x.  pi )  e.  CC
2016, 19mulcli 9604 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
2120a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
22 ine0 9999 . . . . . . . . . 10  |-  _i  =/=  0
23 2ne0 10635 . . . . . . . . . . 11  |-  2  =/=  0
24 pire 22829 . . . . . . . . . . . 12  |-  pi  e.  RR
25 pipos 22831 . . . . . . . . . . . 12  |-  0  <  pi
2624, 25gt0ne0ii 10096 . . . . . . . . . . 11  |-  pi  =/=  0
2717, 18, 23, 26mulne0i 10199 . . . . . . . . . 10  |-  ( 2  x.  pi )  =/=  0
2816, 19, 22, 27mulne0i 10199 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
2928a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
3012, 21, 29divcan2d 10329 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( A  -  ( log `  ( exp `  A
) ) ) )
3130oveq2d 6297 . . . . . 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 9833 . . . . . . 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 2485 . . . . 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 6289 . . . . . . . 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 6297 . . . . . . 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 2457 . . . . . 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 3196 . . . . 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 1018 . . 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 5856 . . . . . 6  |-  ( ( exp `  A )  =  B  ->  ( log `  ( exp `  A
) )  =  ( log `  B ) )
4241oveq1d 6296 . . . . 5  |-  ( ( exp `  A )  =  B  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
4342eqeq2d 2457 . . . 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 2954 . . 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 22934 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
47463adant1 1015 . . . . . . 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 10876 . . . . . . . 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 9579 . . . . . . 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 13811 . . . . . 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 22942 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( exp `  ( log `  B ) )  =  B )
56553adant1 1015 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( exp `  ( log `  B
) )  =  B )
57 ef2kpi 22849 . . . . . 6  |-  ( n  e.  ZZ  ->  ( exp `  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  1 )
5856, 57oveqan12d 6300 . . . . 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 1001 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  B  e.  CC )
6059mulid1d 9616 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( B  x.  1 )  =  B )
6154, 58, 603eqtrd 2488 . . . 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 5856 . . . . 5  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( exp `  A )  =  ( exp `  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
6362eqeq1d 2445 . . . 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 2935 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   _ici 9497    + caddc 9498    x. cmul 9500    - cmin 9810    / cdiv 10213   2c2 10592   ZZcz 10871   expce 13779   picpi 13784   logclog 22920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-sin 13787  df-cos 13788  df-pi 13790  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-limc 22248  df-dv 22249  df-log 22922
This theorem is referenced by:  cxpeq  23109
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