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Theorem cjexp 10870
Description: Complex conjugate of positive integer exponentiation. (Contributed by NM, 7-Jun-2006.)
Assertion
Ref Expression
cjexp  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( * `  ( A ^ N ) )  =  ( ( * `
 A ) ^ N ) )

Proof of Theorem cjexp
Dummy variables  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5873 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5511 . . . . 5  |-  ( j  =  0  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ 0 ) ) )
3 oveq2 5873 . . . . 5  |-  ( j  =  0  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
0 ) )
42, 3eqeq12d 2190 . . . 4  |-  ( j  =  0  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) )
54imbi2d 230 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) ) )
6 oveq2 5873 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5511 . . . . 5  |-  ( j  =  k  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ k ) ) )
8 oveq2 5873 . . . . 5  |-  ( j  =  k  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
k ) )
97, 8eqeq12d 2190 . . . 4  |-  ( j  =  k  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) )
109imbi2d 230 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) ) )
11 oveq2 5873 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5511 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5873 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2190 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) )
1514imbi2d 230 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) ) )
16 oveq2 5873 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5511 . . . . 5  |-  ( j  =  N  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ N ) ) )
18 oveq2 5873 . . . . 5  |-  ( j  =  N  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^ N ) )
1917, 18eqeq12d 2190 . . . 4  |-  ( j  =  N  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) )
2019imbi2d 230 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) ) )
21 exp0 10494 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2221fveq2d 5511 . . . 4  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( * ` 
1 ) )
23 cjcl 10825 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
24 exp0 10494 . . . . . 6  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  1 )
25 1re 7931 . . . . . . 7  |-  1  e.  RR
26 cjre 10859 . . . . . . 7  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
2725, 26ax-mp 5 . . . . . 6  |-  ( * `
 1 )  =  1
2824, 27eqtr4di 2226 . . . . 5  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
2923, 28syl 14 . . . 4  |-  ( A  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
3022, 29eqtr4d 2211 . . 3  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( ( * `
 A ) ^
0 ) )
31 expp1 10497 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3231fveq2d 5511 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( * `  ( ( A ^
k )  x.  A
) ) )
33 expcl 10508 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
34 simpl 109 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
35 cjmul 10862 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3633, 34, 35syl2anc 411 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3732, 36eqtrd 2208 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
3837adantr 276 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
39 oveq1 5872 . . . . . . . 8  |-  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
( * `  ( A ^ k ) )  x.  ( * `  A ) )  =  ( ( ( * `
 A ) ^
k )  x.  (
* `  A )
) )
40 expp1 10497 . . . . . . . . . 10  |-  ( ( ( * `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4123, 40sylan 283 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4241eqcomd 2181 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( * `
 A ) ^
k )  x.  (
* `  A )
)  =  ( ( * `  A ) ^ ( k  +  1 ) ) )
4339, 42sylan9eqr 2230 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( ( * `  ( A ^ k ) )  x.  ( * `
 A ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4438, 43eqtrd 2208 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4544exp31 364 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
* `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
4645com12 30 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  CC  ->  (
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
* `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
4746a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  CC  ->  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( A  e.  CC  ->  ( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
485, 10, 15, 20, 30, 47nn0ind 9340 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  (
* `  ( A ^ N ) )  =  ( ( * `  A ) ^ N
) ) )
4948impcom 125 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( * `  ( A ^ N ) )  =  ( ( * `
 A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    x. cmul 7791   NN0cn0 9149   ^cexp 10489   *ccj 10816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-n0 9150  df-z 9227  df-uz 9502  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821
This theorem is referenced by:  cjexpd  10935  efcj  11649
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