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Theorem efcj 12247
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 11467 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
2 eqid 2253 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) )
32efcvg 12240 . . 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 10141 . . 3  |-  NN0  =  ( ZZ>= `  0 )
6 eqid 2253 . . . 4  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
76efcvg 12240 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
8 seqex 10926 . . . 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 9914 . . . 4  |-  0  e.  ZZ
1110a1i 12 . . 3  |-  ( A  e.  CC  ->  0  e.  ZZ )
126eftval 12232 . . . . . . 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 12229 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1513, 14eqeltrd 2327 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
165, 11, 15serf 10952 . . . 4  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) : NN0 --> CC )
17 ffvelrn 5515 . . . 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 8699 . . . . . 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 10700 . . . . . 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 2343 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
26 cjadd 11503 . . . . . 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 10999 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
29 faccl 11176 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3029adantl 454 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  NN )
3130nncnd 9642 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  CC )
3230nnne0d 9670 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  =/=  0 )
3328, 31, 32cjdivd 11585 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( * `  ( A ^ k ) )  /  ( * `  ( ! `  k ) ) ) )
34 cjexp 11512 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )
3530nnred 9641 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  RR )
3635cjred 11588 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( ! `  k )
)  =  ( ! `
 k ) )
3734, 36oveq12d 5728 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  ( A ^ k ) )  /  ( * `
 ( ! `  k ) ) )  =  ( ( ( * `  A ) ^ k )  / 
( ! `  k
) ) )
3833, 37eqtrd 2285 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( ( * `  A
) ^ k )  /  ( ! `  k ) ) )
3913fveq2d 5381 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( * `  ( ( A ^ k )  /  ( ! `  k ) ) ) )
402eftval 12232 . . . . . . . 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 2295 . . . . . 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 10971 . . . 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 2258 . . 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 11955 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( * `
 ( exp `  A
) ) )
47 climuni 11903 . 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 2727   class class class wbr 3920    e. cmpt 3974   -->wf 4588   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617    + caddc 8620    / cdiv 9303   NNcn 9626   NN0cn0 9844   ZZcz 9903   ZZ>=cuz 10109   ...cfz 10660    seq cseq 10924   ^cexp 10982   !cfa 11166   *ccj 11458    ~~> cli 11835   expce 12217
This theorem is referenced by:  resinval  12289  recosval  12290  logcj  19792  cosargd  19794
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-fac 11167  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223
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