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Theorem efcj 12369
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 11586 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
2 eqid 2284 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( * `  A
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( * `  A ) ^ n )  / 
( ! `  n
) ) )
32efcvg 12362 . . 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 10258 . . 3  |-  NN0  =  ( ZZ>= `  0 )
6 eqid 2284 . . . 4  |-  ( n  e.  NN0  |->  ( ( A ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) )
76efcvg 12362 . . 3  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) )  ~~>  ( exp `  A ) )
8 seqex 11044 . . . 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 10031 . . . 4  |-  0  e.  ZZ
1110a1i 10 . . 3  |-  ( A  e.  CC  ->  0  e.  ZZ )
126eftval 12354 . . . . . . 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 12351 . . . . . 6  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( A ^
k )  /  ( ! `  k )
)  e.  CC )
1513, 14eqeltrd 2358 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( n  e. 
NN0  |->  ( ( A ^ n )  / 
( ! `  n
) ) ) `  k )  e.  CC )
165, 11, 15serf 11070 . . . 4  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) ) : NN0 --> CC )
17 ffvelrn 5625 . . . 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 8815 . . . . . 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 10818 . . . . . 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 2374 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
j  e.  ( ZZ>= ` 
0 ) )
26 cjadd 11622 . . . . . 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 11117 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
29 faccl 11294 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ! `
 k )  e.  NN )
3029adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  NN )
3130nncnd 9758 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  CC )
3230nnne0d 9786 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  =/=  0 )
3328, 31, 32cjdivd 11704 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( * `  ( A ^ k ) )  /  ( * `  ( ! `  k ) ) ) )
34 cjexp 11631 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )
3530nnred 9757 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ! `  k
)  e.  RR )
3635cjred 11707 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( ! `  k )
)  =  ( ! `
 k ) )
3734, 36oveq12d 5838 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  ( A ^ k ) )  /  ( * `
 ( ! `  k ) ) )  =  ( ( ( * `  A ) ^ k )  / 
( ! `  k
) ) )
3833, 37eqtrd 2316 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  /  ( ! `
 k ) ) )  =  ( ( ( * `  A
) ^ k )  /  ( ! `  k ) ) )
3913fveq2d 5490 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) ) `  k
) )  =  ( * `  ( ( A ^ k )  /  ( ! `  k ) ) ) )
402eftval 12354 . . . . . . . 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 2326 . . . . . 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 11089 . . . 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 2289 . . 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 12074 . 2  |-  ( A  e.  CC  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( ( ( * `
 A ) ^
n )  /  ( ! `  n )
) ) )  ~~>  ( * `
 ( exp `  A
) ) )
47 climuni 12022 . 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 1623    e. wcel 1685   _Vcvv 2789   class class class wbr 4024    e. cmpt 4078   -->wf 5217   ` cfv 5221  (class class class)co 5820   CCcc 8731   0cc0 8733    + caddc 8736    / cdiv 9419   NNcn 9742   NN0cn0 9961   ZZcz 10020   ZZ>=cuz 10226   ...cfz 10778    seq cseq 11042   ^cexp 11100   !cfa 11284   *ccj 11577    ~~> cli 11954   expce 12339
This theorem is referenced by:  resinval  12411  recosval  12412  logcj  19956  cosargd  19958
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-pm 6771  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10658  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-fac 11285  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345
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