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Theorem efcj 12300
Description: Exponential function of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
Assertion
Ref Expression
efcj  |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A ) ) )

Proof of Theorem efcj
StepHypRef Expression
1 cjcl 11520 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
2 eqid 2256 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) )
32efcvg 12293 . . 3  |-  ( ( * `  A )  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  ( * `  A
) ) )
41, 3syl 17 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  ( * `  A
) ) )
5 nn0uz 10194 . . 3  |-  NN0  =  ( ZZ>= `  0 )
6 eqid 2256 . . . 4  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
76efcvg 12293 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
8 seqex 10979 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) )  e.  _V
98a1i 12 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  e. 
_V )
10 0z 9967 . . . 4  |-  0  e.  ZZ
1110a1i 12 . . 3  |-  ( A  e.  CC  ->  0  e.  ZZ )
126eftval 12285 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
1312adantl 454 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
14 eftcl 12282 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1513, 14eqeltrd 2330 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
165, 11, 15serf 11005 . . . 4  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) : NN0 --> CC )
17 ffvelrn 5562 . . . 4  |-  ( (  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) : NN0 --> CC  /\  j  e.  NN0 )  -> 
(  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j )  e.  CC )
1816, 17sylan 459 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
(  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j )  e.  CC )
19 addcl 8752 . . . . . 6  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
2019adantl 454 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  ( k  e.  CC  /\  m  e.  CC ) )  ->  ( k  +  m )  e.  CC )
21 simpl 445 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  ->  A  e.  CC )
22 elfznn0 10753 . . . . . 6  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
2321, 22, 15syl2an 465 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  CC )
24 simpr 449 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  NN0 )
2524, 5syl6eleq 2346 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
26 cjadd 11556 . . . . . 6  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( * `  (
k  +  m ) )  =  ( ( * `  k )  +  ( * `  m ) ) )
2726adantl 454 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  ( k  e.  CC  /\  m  e.  CC ) )  ->  ( * `  ( k  +  m
) )  =  ( ( * `  k
)  +  ( * `
 m ) ) )
28 expcl 11052 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
29 faccl 11229 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3029adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  NN )
3130nncnd 9695 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  CC )
3230nnne0d 9723 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  =/=  0 )
3328, 31, 32cjdivd 11638 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( * `  ( A ^ k ) )  /  ( * `  ( ! `  k ) ) ) )
34 cjexp 11565 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )
3530nnred 9694 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  RR )
3635cjred 11641 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( ! `  k )
)  =  ( ! `
 k ) )
3734, 36oveq12d 5775 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  ( A ^ k ) )  /  ( * `
 ( ! `  k ) ) )  =  ( ( ( * `  A ) ^ k )  / 
( ! `  k
) ) )
3833, 37eqtrd 2288 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( ( * `  A
) ^ k )  /  ( ! `  k ) ) )
3913fveq2d 5427 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( * `  ( ( A ^ k )  /  ( ! `  k ) ) ) )
402eftval 12285 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( ( * `
 A ) ^
k )  /  ( ! `  k )
) )
4140adantl 454 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( ( * `  A ) ^ k
)  /  ( ! `
 k ) ) )
4238, 39, 413eqtr4d 2298 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( ( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) `  k
) )
4321, 22, 42syl2an 465 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( * `  ( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k ) )  =  ( ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) ) `  k ) )
4420, 23, 25, 27, 43seqhomo 11024 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( * `  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) `  j ) )  =  (  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) ) `  j ) )
4544eqcomd 2261 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
(  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) ) `  j )  =  ( * `  (  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) ) `
 j ) ) )
465, 7, 9, 11, 18, 45climcj 12008 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( * `
 ( exp `  A
) ) )
47 climuni 11956 . 2  |-  ( (  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) ) )  ~~>  ( exp `  (
* `  A )
)  /\  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) )  ~~>  ( * `  ( exp `  A ) ) )  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A ) ) )
484, 46, 47syl2anc 645 1  |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740   class class class wbr 3963    e. cmpt 4017   -->wf 4634   ` cfv 4638  (class class class)co 5757   CCcc 8668   0cc0 8670    + caddc 8673    / cdiv 9356   NNcn 9679   NN0cn0 9897   ZZcz 9956   ZZ>=cuz 10162   ...cfz 10713    seq cseq 10977   ^cexp 11035   !cfa 11219   *ccj 11511    ~~> cli 11888   expce 12270
This theorem is referenced by:  resinval  12342  recosval  12343  logcj  19887  cosargd  19889
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-pm 6708  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-ico 10593  df-fz 10714  df-fzo 10802  df-fl 10856  df-seq 10978  df-exp 11036  df-fac 11220  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276
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