MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eflogeq Structured version   Unicode version

Theorem eflogeq 22049
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 13367 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efne0 13380 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
31, 2logcld 22021 . . . . . . . 8  |-  ( A  e.  CC  ->  ( log `  ( exp `  A
) )  e.  CC )
4 efsub 13383 . . . . . . . 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 22027 . . . . . . . . 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 6106 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A
)  /  ( exp `  A ) ) )
91, 2dividd 10104 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  A ) )  =  1 )
105, 8, 93eqtrd 2478 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1 )
11 subcl 9608 . . . . . . . 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 21984 . . . . . . 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 9340 . . . . . . . . . 10  |-  _i  e.  CC
17 2cn 10391 . . . . . . . . . . 11  |-  2  e.  CC
18 picn 21921 . . . . . . . . . . 11  |-  pi  e.  CC
1917, 18mulcli 9390 . . . . . . . . . 10  |-  ( 2  x.  pi )  e.  CC
2016, 19mulcli 9390 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
2120a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
22 ine0 9779 . . . . . . . . . 10  |-  _i  =/=  0
23 2ne0 10413 . . . . . . . . . . 11  |-  2  =/=  0
24 pire 21920 . . . . . . . . . . . 12  |-  pi  e.  RR
25 pipos 21922 . . . . . . . . . . . 12  |-  0  <  pi
2624, 25gt0ne0ii 9875 . . . . . . . . . . 11  |-  pi  =/=  0
2717, 18, 23, 26mulne0i 9978 . . . . . . . . . 10  |-  ( 2  x.  pi )  =/=  0
2816, 19, 22, 27mulne0i 9978 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
2928a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
3012, 21, 29divcan2d 10108 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( A  -  ( log `  ( exp `  A
) ) ) )
3130oveq2d 6106 . . . . . 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 9617 . . . . . . 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 2475 . . . . 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 6098 . . . . . . . 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 6106 . . . . . . 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 2453 . . . . . 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 3072 . . . . 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 1009 . . 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 5690 . . . . . 6  |-  ( ( exp `  A )  =  B  ->  ( log `  ( exp `  A
) )  =  ( log `  B ) )
4241oveq1d 6105 . . . . 5  |-  ( ( exp `  A )  =  B  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
4342eqeq2d 2453 . . . 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 2735 . . 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 22019 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
47463adant1 1006 . . . . . . 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 10650 . . . . . . . 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 9365 . . . . . . 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 13378 . . . . . 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 22027 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( exp `  ( log `  B ) )  =  B )
56553adant1 1006 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( exp `  ( log `  B
) )  =  B )
57 ef2kpi 21939 . . . . . 6  |-  ( n  e.  ZZ  ->  ( exp `  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  1 )
5856, 57oveqan12d 6109 . . . . 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 992 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  B  e.  CC )
6059mulid1d 9402 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( B  x.  1 )  =  B )
6154, 58, 603eqtrd 2478 . . . 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 5690 . . . . 5  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( exp `  A )  =  ( exp `  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
6362eqeq1d 2450 . . . 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 2840 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715   ` cfv 5417  (class class class)co 6090   CCcc 9279   0cc0 9281   1c1 9282   _ici 9283    + caddc 9284    x. cmul 9286    - cmin 9594    / cdiv 9992   2c2 10370   ZZcz 10645   expce 13346   picpi 13351   logclog 22005
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-fi 7660  df-sup 7690  df-oi 7723  df-card 8108  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-ioo 11303  df-ioc 11304  df-ico 11305  df-icc 11306  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-shft 12555  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-limsup 12948  df-clim 12965  df-rlim 12966  df-sum 13163  df-ef 13352  df-sin 13354  df-cos 13355  df-pi 13357  df-struct 14175  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-mulr 14251  df-starv 14252  df-sca 14253  df-vsca 14254  df-ip 14255  df-tset 14256  df-ple 14257  df-ds 14259  df-unif 14260  df-hom 14261  df-cco 14262  df-rest 14360  df-topn 14361  df-0g 14379  df-gsum 14380  df-topgen 14381  df-pt 14382  df-prds 14385  df-xrs 14439  df-qtop 14444  df-imas 14445  df-xps 14447  df-mre 14523  df-mrc 14524  df-acs 14526  df-mnd 15414  df-submnd 15464  df-mulg 15547  df-cntz 15834  df-cmn 16278  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-fbas 17813  df-fg 17814  df-cnfld 17818  df-top 18502  df-bases 18504  df-topon 18505  df-topsp 18506  df-cld 18622  df-ntr 18623  df-cls 18624  df-nei 18701  df-lp 18739  df-perf 18740  df-cn 18830  df-cnp 18831  df-haus 18918  df-tx 19134  df-hmeo 19327  df-fil 19418  df-fm 19510  df-flim 19511  df-flf 19512  df-xms 19894  df-ms 19895  df-tms 19896  df-cncf 20453  df-limc 21340  df-dv 21341  df-log 22007
This theorem is referenced by:  cxpeq  22194
  Copyright terms: Public domain W3C validator