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Theorem eflogeq 19955
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 12364 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efne0 12377 . . . . . . . . 9  |-  ( A  e.  CC  ->  ( exp `  A )  =/=  0 )
3 logcl 19926 . . . . . . . . 9  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A )  =/=  0 )  -> 
( log `  ( exp `  A ) )  e.  CC )
41, 2, 3syl2anc 642 . . . . . . . 8  |-  ( A  e.  CC  ->  ( log `  ( exp `  A
) )  e.  CC )
5 efsub 12380 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
64, 5mpdan 649 . . . . . . 7  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A )  /  ( exp `  ( log `  ( exp `  A
) ) ) ) )
7 eflog 19933 . . . . . . . . 9  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  A )  =/=  0 )  -> 
( exp `  ( log `  ( exp `  A
) ) )  =  ( exp `  A
) )
81, 2, 7syl2anc 642 . . . . . . . 8  |-  ( A  e.  CC  ->  ( exp `  ( log `  ( exp `  A ) ) )  =  ( exp `  A ) )
98oveq2d 5874 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  ( log `  ( exp `  A ) ) ) )  =  ( ( exp `  A
)  /  ( exp `  A ) ) )
101, 2dividd 9534 . . . . . . 7  |-  ( A  e.  CC  ->  (
( exp `  A
)  /  ( exp `  A ) )  =  1 )
116, 9, 103eqtrd 2319 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1 )
12 subcl 9051 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( log `  ( exp `  A ) )  e.  CC )  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
134, 12mpdan 649 . . . . . . 7  |-  ( A  e.  CC  ->  ( A  -  ( log `  ( exp `  A
) ) )  e.  CC )
14 efeq1 19891 . . . . . . 7  |-  ( ( A  -  ( log `  ( exp `  A
) ) )  e.  CC  ->  ( ( exp `  ( A  -  ( log `  ( exp `  A ) ) ) )  =  1  <->  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
1513, 14syl 15 . . . . . 6  |-  ( A  e.  CC  ->  (
( exp `  ( A  -  ( log `  ( exp `  A
) ) ) )  =  1  <->  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ ) )
1611, 15mpbid 201 . . . . 5  |-  ( A  e.  CC  ->  (
( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) )  e.  ZZ )
17 ax-icn 8796 . . . . . . . . . 10  |-  _i  e.  CC
18 2cn 9816 . . . . . . . . . . 11  |-  2  e.  CC
19 pire 19832 . . . . . . . . . . . 12  |-  pi  e.  RR
2019recni 8849 . . . . . . . . . . 11  |-  pi  e.  CC
2118, 20mulcli 8842 . . . . . . . . . 10  |-  ( 2  x.  pi )  e.  CC
2217, 21mulcli 8842 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
2322a1i 10 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  e.  CC )
24 ine0 9215 . . . . . . . . . 10  |-  _i  =/=  0
25 2ne0 9829 . . . . . . . . . . 11  |-  2  =/=  0
26 pipos 19833 . . . . . . . . . . . 12  |-  0  <  pi
2719, 26gt0ne0ii 9309 . . . . . . . . . . 11  |-  pi  =/=  0
2818, 20, 25, 27mulne0i 9411 . . . . . . . . . 10  |-  ( 2  x.  pi )  =/=  0
2917, 21, 24, 28mulne0i 9411 . . . . . . . . 9  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
3029a1i 10 . . . . . . . 8  |-  ( A  e.  CC  ->  (
_i  x.  ( 2  x.  pi ) )  =/=  0 )
3113, 23, 30divcan2d 9538 . . . . . . 7  |-  ( A  e.  CC  ->  (
( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) )  =  ( A  -  ( log `  ( exp `  A
) ) ) )
3231oveq2d 5874 . . . . . 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
) ) ) ) )
33 pncan3 9059 . . . . . . 7  |-  ( ( ( log `  ( exp `  A ) )  e.  CC  /\  A  e.  CC )  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
344, 33mpancom 650 . . . . . 6  |-  ( A  e.  CC  ->  (
( log `  ( exp `  A ) )  +  ( A  -  ( log `  ( exp `  A ) ) ) )  =  A )
3532, 34eqtr2d 2316 . . . . 5  |-  ( A  e.  CC  ->  A  =  ( ( log `  ( exp `  A
) )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  ( ( A  -  ( log `  ( exp `  A
) ) )  / 
( _i  x.  (
2  x.  pi ) ) ) ) ) )
36 oveq2 5866 . . . . . . . 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 ) ) ) ) )
3736oveq2d 5874 . . . . . . 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 ) ) ) ) ) )
3837eqeq2d 2294 . . . . . 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 ) ) ) ) ) ) )
3938rspcev 2884 . . . . 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 ) ) )
4016, 35, 39syl2anc 642 . . . 4  |-  ( A  e.  CC  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
41403ad2ant1 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  E. n  e.  ZZ  A  =  ( ( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
42 fveq2 5525 . . . . . 6  |-  ( ( exp `  A )  =  B  ->  ( log `  ( exp `  A
) )  =  ( log `  B ) )
4342oveq1d 5873 . . . . 5  |-  ( ( exp `  A )  =  B  ->  (
( log `  ( exp `  A ) )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  ( ( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )
4443eqeq2d 2294 . . . 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 ) ) ) )
4544rexbidv 2564 . . 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
) ) ) )
4641, 45syl5ibcom 211 . 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
) ) ) )
47 logcl 19926 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( log `  B
)  e.  CC )
48473adant1 973 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( log `  B )  e.  CC )
4948adantr 451 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( log `  B
)  e.  CC )
50 zcn 10029 . . . . . . . 8  |-  ( n  e.  ZZ  ->  n  e.  CC )
5150adantl 452 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  n  e.  CC )
52 mulcl 8821 . . . . . . 7  |-  ( ( ( _i  x.  (
2  x.  pi ) )  e.  CC  /\  n  e.  CC )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
5322, 51, 52sylancr 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( ( _i  x.  ( 2  x.  pi ) )  x.  n
)  e.  CC )
54 efadd 12375 . . . . . 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 ) ) ) )
5549, 53, 54syl2anc 642 . . . . 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 ) ) ) )
56 eflog 19933 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( exp `  ( log `  B ) )  =  B )
57563adant1 973 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( exp `  ( log `  B
) )  =  B )
58 ef2kpi 19846 . . . . . 6  |-  ( n  e.  ZZ  ->  ( exp `  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) )  =  1 )
5957, 58oveqan12d 5877 . . . . 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 ) )
60 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  B  e.  CC )
6160mulid1d 8852 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( B  x.  1 )  =  B )
6255, 59, 613eqtrd 2319 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  /\  n  e.  ZZ )  ->  ( exp `  (
( log `  B
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) )  =  B )
63 fveq2 5525 . . . . 5  |-  ( A  =  ( ( log `  B )  +  ( ( _i  x.  (
2  x.  pi ) )  x.  n ) )  ->  ( exp `  A )  =  ( exp `  ( ( log `  B )  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  n ) ) ) )
6463eqeq1d 2291 . . . 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 ) )
6562, 64syl5ibrcom 213 . . 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 ) )
6665rexlimdva 2667 . 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 ) )
6746, 66impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037    / cdiv 9423   2c2 9795   ZZcz 10024   expce 12343   picpi 12348   logclog 19912
This theorem is referenced by:  cxpeq  20097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914
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