ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cjexp Unicode version

Theorem cjexp 10934
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 5904 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5538 . . . . 5  |-  ( j  =  0  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ 0 ) ) )
3 oveq2 5904 . . . . 5  |-  ( j  =  0  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
0 ) )
42, 3eqeq12d 2204 . . . 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 5904 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5538 . . . . 5  |-  ( j  =  k  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ k ) ) )
8 oveq2 5904 . . . . 5  |-  ( j  =  k  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
k ) )
97, 8eqeq12d 2204 . . . 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 5904 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5538 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5904 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2204 . . . 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 5904 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5538 . . . . 5  |-  ( j  =  N  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ N ) ) )
18 oveq2 5904 . . . . 5  |-  ( j  =  N  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^ N ) )
1917, 18eqeq12d 2204 . . . 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 10555 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2221fveq2d 5538 . . . 4  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( * ` 
1 ) )
23 cjcl 10889 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
24 exp0 10555 . . . . . 6  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  1 )
25 1re 7986 . . . . . . 7  |-  1  e.  RR
26 cjre 10923 . . . . . . 7  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
2725, 26ax-mp 5 . . . . . 6  |-  ( * `
 1 )  =  1
2824, 27eqtr4di 2240 . . . . 5  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
2923, 28syl 14 . . . 4  |-  ( A  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
3022, 29eqtr4d 2225 . . 3  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( ( * `
 A ) ^
0 ) )
31 expp1 10558 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3231fveq2d 5538 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( * `  ( ( A ^
k )  x.  A
) ) )
33 expcl 10569 . . . . . . . . . 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 10926 . . . . . . . . . 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 2222 . . . . . . . 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 5903 . . . . . . . 8  |-  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
( * `  ( A ^ k ) )  x.  ( * `  A ) )  =  ( ( ( * `
 A ) ^
k )  x.  (
* `  A )
) )
40 expp1 10558 . . . . . . . . . 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 2195 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( * `
 A ) ^
k )  x.  (
* `  A )
)  =  ( ( * `  A ) ^ ( k  +  1 ) ) )
4339, 42sylan9eqr 2244 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( ( * `  ( A ^ k ) )  x.  ( * `
 A ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4438, 43eqtrd 2222 . . . . . 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 9397 . 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 1364    e. wcel 2160   ` cfv 5235  (class class class)co 5896   CCcc 7839   RRcr 7840   0cc0 7841   1c1 7842    + caddc 7844    x. cmul 7846   NN0cn0 9206   ^cexp 10550   *ccj 10880
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-n0 9207  df-z 9284  df-uz 9559  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885
This theorem is referenced by:  cjexpd  10999  efcj  11713
  Copyright terms: Public domain W3C validator