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Theorem efcj 12581
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
Dummy variables  j 
k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cjcl 11797 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
2 eqid 2366 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) )
32efcvg 12574 . . 3  |-  ( ( * `  A )  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  ( * `  A
) ) )
41, 3syl 15 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  ( * `  A
) ) )
5 nn0uz 10413 . . 3  |-  NN0  =  ( ZZ>= `  0 )
6 eqid 2366 . . . 4  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
76efcvg 12574 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
8 seqex 11212 . . . 4  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) )  e.  _V
98a1i 10 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  e. 
_V )
10 0z 10186 . . . 4  |-  0  e.  ZZ
1110a1i 10 . . 3  |-  ( A  e.  CC  ->  0  e.  ZZ )
126eftval 12566 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( A ^
k )  /  ( ! `  k )
) )
1312adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( A ^ k
)  /  ( ! `
 k ) ) )
14 eftcl 12563 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1513, 14eqeltrd 2440 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
165, 11, 15serf 11238 . . . 4  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) : NN0 --> CC )
17 ffvelrn 5770 . . . 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 457 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
(  seq  0 (  +  ,  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) ) `  j )  e.  CC )
19 addcl 8966 . . . . . 6  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( k  +  m
)  e.  CC )
2019adantl 452 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  ( k  e.  CC  /\  m  e.  CC ) )  ->  ( k  +  m )  e.  CC )
21 simpl 443 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  ->  A  e.  CC )
22 elfznn0 10975 . . . . . 6  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
2321, 22, 15syl2an 463 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  k  e.  (
0 ... j ) )  ->  ( ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) ) `
 k )  e.  CC )
24 simpr 447 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  NN0 )
2524, 5syl6eleq 2456 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
26 cjadd 11833 . . . . . 6  |-  ( ( k  e.  CC  /\  m  e.  CC )  ->  ( * `  (
k  +  m ) )  =  ( ( * `  k )  +  ( * `  m ) ) )
2726adantl 452 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN0 )  /\  ( k  e.  CC  /\  m  e.  CC ) )  ->  ( * `  ( k  +  m
) )  =  ( ( * `  k
)  +  ( * `
 m ) ) )
28 expcl 11286 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
29 faccl 11463 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3029adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  NN )
3130nncnd 9909 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  CC )
3230nnne0d 9937 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  =/=  0 )
3328, 31, 32cjdivd 11915 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( * `  ( A ^ k ) )  /  ( * `  ( ! `  k ) ) ) )
34 cjexp 11842 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )
3530nnred 9908 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  RR )
3635cjred 11918 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( ! `  k )
)  =  ( ! `
 k ) )
3734, 36oveq12d 5999 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  ( A ^ k ) )  /  ( * `
 ( ! `  k ) ) )  =  ( ( ( * `  A ) ^ k )  / 
( ! `  k
) ) )
3833, 37eqtrd 2398 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( ( * `  A
) ^ k )  /  ( ! `  k ) ) )
3913fveq2d 5636 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( * `  ( ( A ^ k )  /  ( ! `  k ) ) ) )
402eftval 12566 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) ) `
 k )  =  ( ( ( * `
 A ) ^
k )  /  ( ! `  k )
) )
4140adantl 452 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) ) `  k )  =  ( ( ( * `  A ) ^ k
)  /  ( ! `
 k ) ) )
4238, 39, 413eqtr4d 2408 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( ( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) `  k
) )
4321, 22, 42syl2an 463 . . . . 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 11257 . . . 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 2371 . . 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 12285 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( * `
 ( exp `  A
) ) )
47 climuni 12233 . 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 642 1  |-  ( A  e.  CC  ->  ( exp `  ( * `  A ) )  =  ( * `  ( exp `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873   class class class wbr 4125    e. cmpt 4179   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   0cc0 8884    + caddc 8887    / cdiv 9570   NNcn 9893   NN0cn0 10114   ZZcz 10175   ZZ>=cuz 10381   ...cfz 10935    seq cseq 11210   ^cexp 11269   !cfa 11453   *ccj 11788    ~~> cli 12165   expce 12551
This theorem is referenced by:  resinval  12623  recosval  12624  logcj  20179  cosargd  20181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-ico 10815  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-fac 11454  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557
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