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Theorem cjexp 11635
Description: Complex conjugate of natural number 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 5866 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5529 . . . . 5  |-  ( j  =  0  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ 0 ) ) )
3 oveq2 5866 . . . . 5  |-  ( j  =  0  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
0 ) )
42, 3eqeq12d 2297 . . . 4  |-  ( j  =  0  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) )
54imbi2d 307 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) ) )
6 oveq2 5866 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5529 . . . . 5  |-  ( j  =  k  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ k ) ) )
8 oveq2 5866 . . . . 5  |-  ( j  =  k  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
k ) )
97, 8eqeq12d 2297 . . . 4  |-  ( j  =  k  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) )
109imbi2d 307 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) ) )
11 oveq2 5866 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5529 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5866 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2297 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) )
1514imbi2d 307 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) ) )
16 oveq2 5866 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5529 . . . . 5  |-  ( j  =  N  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ N ) ) )
18 oveq2 5866 . . . . 5  |-  ( j  =  N  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^ N ) )
1917, 18eqeq12d 2297 . . . 4  |-  ( j  =  N  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) )
2019imbi2d 307 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) ) )
21 exp0 11108 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2221fveq2d 5529 . . . 4  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( * ` 
1 ) )
23 cjcl 11590 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
24 exp0 11108 . . . . . 6  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  1 )
25 1re 8837 . . . . . . 7  |-  1  e.  RR
26 cjre 11624 . . . . . . 7  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
2725, 26ax-mp 8 . . . . . 6  |-  ( * `
 1 )  =  1
2824, 27syl6eqr 2333 . . . . 5  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
2923, 28syl 15 . . . 4  |-  ( A  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
3022, 29eqtr4d 2318 . . 3  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( ( * `
 A ) ^
0 ) )
31 expp1 11110 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3231fveq2d 5529 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( * `  ( ( A ^
k )  x.  A
) ) )
33 expcl 11121 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
34 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
35 cjmul 11627 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3633, 34, 35syl2anc 642 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3732, 36eqtrd 2315 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
3837adantr 451 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
39 oveq1 5865 . . . . . . . 8  |-  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
( * `  ( A ^ k ) )  x.  ( * `  A ) )  =  ( ( ( * `
 A ) ^
k )  x.  (
* `  A )
) )
40 expp1 11110 . . . . . . . . . 10  |-  ( ( ( * `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4123, 40sylan 457 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4241eqcomd 2288 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( * `
 A ) ^
k )  x.  (
* `  A )
)  =  ( ( * `  A ) ^ ( k  +  1 ) ) )
4339, 42sylan9eqr 2337 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( ( * `  ( A ^ k ) )  x.  ( * `
 A ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4438, 43eqtrd 2315 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4544exp31 587 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
* `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
4645com12 27 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  CC  ->  (
( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
* `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
4746a2d 23 . . 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 10108 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  (
* `  ( A ^ N ) )  =  ( ( * `  A ) ^ N
) ) )
4948impcom 419 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( * `  ( A ^ N ) )  =  ( ( * `
 A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NN0cn0 9965   ^cexp 11104   *ccj 11581
This theorem is referenced by:  cjexpd  11698  efcj  12373  plycjlem  19657  plyrecj  19660  atandmcj  20205
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586
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