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

Theorem eflogeq 20496
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 12685 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efne0 12698 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
31, 2logcld 20468 . . . . . . . 8  |-  ( A  e.  CC  ->  ( log `  ( exp `  A
) )  e.  CC )
4 efsub 12701 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
53, 4mpdan 650 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
6 eflog 20474 . . . . . . . . 9  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A )  =/=  0 )  -> 
( exp `  ( log `  ( exp `  A
) ) )  =  ( exp `  A
) )
71, 2, 6syl2anc 643 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( log `  ( exp `  A ) ) )  =  ( exp `  A ) )
87oveq2d 6097 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A
)  /  ( exp `  A ) ) )
91, 2dividd 9788 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  A ) )  =  1 )
105, 8, 93eqtrd 2472 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1 )
11 subcl 9305 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
123, 11mpdan 650 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
13 efeq1 20431 . . . . . . 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 202 . . . . 5  |-  ( A  e.  CC  ->  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ )
16 ax-icn 9049 . . . . . . . . . 10  |-  _i  e.  CC
17 2cn 10070 . . . . . . . . . . 11  |-  2  e.  CC
18 pire 20372 . . . . . . . . . . . 12  |-  pi  e.  RR
1918recni 9102 . . . . . . . . . . 11  |-  pi  e.  CC
2017, 19mulcli 9095 . . . . . . . . . 10  |-  ( 2  x.  pi )  e.  CC
2116, 20mulcli 9095 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
2221a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
23 ine0 9469 . . . . . . . . . 10  |-  _i  =/=  0
24 2ne0 10083 . . . . . . . . . . 11  |-  2  =/=  0
25 pipos 20373 . . . . . . . . . . . 12  |-  0  <  pi
2618, 25gt0ne0ii 9563 . . . . . . . . . . 11  |-  pi  =/=  0
2717, 19, 24, 26mulne0i 9665 . . . . . . . . . 10  |-  ( 2  x.  pi )  =/=  0
2816, 20, 23, 27mulne0i 9665 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
2928a1i 11 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
3012, 22, 29divcan2d 9792 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( A  -  ( log `  ( exp `  A
) ) ) )
3130oveq2d 6097 . . . . . 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 9313 . . . . . . 7  |-  ( ( ( log `  ( exp `  A ) )  e.  CC  /\  A  e.  CC )  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
333, 32mpancom 651 . . . . . 6  |-  ( A  e.  CC  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
3431, 33eqtr2d 2469 . . . . 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 6089 . . . . . . . 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 6097 . . . . . . 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 2447 . . . . . 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 3052 . . . . 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 643 . . . 4  |-  ( A  e.  CC  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
40393ad2ant1 978 . . 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 5728 . . . . . 6  |-  ( ( exp `  A )  =  B  ->  ( log `  ( exp `  A
) )  =  ( log `  B ) )
4241oveq1d 6096 . . . . 5  |-  ( ( exp `  A )  =  B  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
4342eqeq2d 2447 . . . 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 2726 . . 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 212 . 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 20466 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
47463adant1 975 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( log `  B )  e.  CC )
4847adantr 452 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( log `  B
)  e.  CC )
49 zcn 10287 . . . . . . . 8  |-  ( n  e.  ZZ  ->  n  e.  CC )
5049adantl 453 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  n  e.  CC )
51 mulcl 9074 . . . . . . 7  |-  ( ( ( _i  x.  (
2  x.  pi ) )  e.  CC  /\  n  e.  CC )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
5221, 50, 51sylancr 645 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
53 efadd 12696 . . . . . 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 643 . . . . 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 20474 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( exp `  ( log `  B ) )  =  B )
56553adant1 975 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( exp `  ( log `  B
) )  =  B )
57 ef2kpi 20386 . . . . . 6  |-  ( n  e.  ZZ  ->  ( exp `  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  1 )
5856, 57oveqan12d 6100 . . . . 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 961 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  B  e.  CC )
6059mulid1d 9105 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( B  x.  1 )  =  B )
6154, 58, 603eqtrd 2472 . . . 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 5728 . . . . 5  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( exp `  A )  =  ( exp `  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
6362eqeq1d 2444 . . . 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 214 . . 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 2830 . 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 184 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    + caddc 8993    x. cmul 8995    - cmin 9291    / cdiv 9677   2c2 10049   ZZcz 10282   expce 12664   picpi 12669   logclog 20452
This theorem is referenced by:  cxpeq  20641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454
  Copyright terms: Public domain W3C validator