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Theorem cjexp 12945
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 6291 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5869 . . . . 5  |-  ( j  =  0  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ 0 ) ) )
3 oveq2 6291 . . . . 5  |-  ( j  =  0  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
0 ) )
42, 3eqeq12d 2489 . . . 4  |-  ( j  =  0  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) )
54imbi2d 316 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) ) )
6 oveq2 6291 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5869 . . . . 5  |-  ( j  =  k  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ k ) ) )
8 oveq2 6291 . . . . 5  |-  ( j  =  k  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
k ) )
97, 8eqeq12d 2489 . . . 4  |-  ( j  =  k  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) )
109imbi2d 316 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) ) )
11 oveq2 6291 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5869 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 6291 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2489 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) )
1514imbi2d 316 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) ) )
16 oveq2 6291 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5869 . . . . 5  |-  ( j  =  N  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ N ) ) )
18 oveq2 6291 . . . . 5  |-  ( j  =  N  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^ N ) )
1917, 18eqeq12d 2489 . . . 4  |-  ( j  =  N  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) )
2019imbi2d 316 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) ) )
21 exp0 12137 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2221fveq2d 5869 . . . 4  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( * ` 
1 ) )
23 cjcl 12900 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
24 exp0 12137 . . . . . 6  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  1 )
25 1re 9594 . . . . . . 7  |-  1  e.  RR
26 cjre 12934 . . . . . . 7  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
2725, 26ax-mp 5 . . . . . 6  |-  ( * `
 1 )  =  1
2824, 27syl6eqr 2526 . . . . 5  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
2923, 28syl 16 . . . 4  |-  ( A  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
3022, 29eqtr4d 2511 . . 3  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( ( * `
 A ) ^
0 ) )
31 expp1 12140 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3231fveq2d 5869 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( * `  ( ( A ^
k )  x.  A
) ) )
33 expcl 12151 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
34 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
35 cjmul 12937 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3633, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3732, 36eqtrd 2508 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
3837adantr 465 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
39 oveq1 6290 . . . . . . . 8  |-  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
( * `  ( A ^ k ) )  x.  ( * `  A ) )  =  ( ( ( * `
 A ) ^
k )  x.  (
* `  A )
) )
40 expp1 12140 . . . . . . . . . 10  |-  ( ( ( * `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4123, 40sylan 471 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4241eqcomd 2475 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( * `
 A ) ^
k )  x.  (
* `  A )
)  =  ( ( * `  A ) ^ ( k  +  1 ) ) )
4339, 42sylan9eqr 2530 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( ( * `  ( A ^ k ) )  x.  ( * `
 A ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4438, 43eqtrd 2508 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4544exp31 604 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  ->  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
* `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) ) ) )
4645com12 31 . . . 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 10956 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  (
* `  ( A ^ N ) )  =  ( ( * `  A ) ^ N
) ) )
4948impcom 430 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( * `  ( A ^ N ) )  =  ( ( * `
 A ) ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5587  (class class class)co 6283   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496   NN0cn0 10794   ^cexp 12133   *ccj 12891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896
This theorem is referenced by:  cjexpd  13008  efcj  13688  plycjlem  22423  plyrecj  22426  atandmcj  22984
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