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

Theorem cjexp 9493
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 5520 . . . . . 6  |-  ( j  =  0  ->  ( A ^ j )  =  ( A ^ 0 ) )
21fveq2d 5182 . . . . 5  |-  ( j  =  0  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ 0 ) ) )
3 oveq2 5520 . . . . 5  |-  ( j  =  0  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
0 ) )
42, 3eqeq12d 2054 . . . 4  |-  ( j  =  0  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) )
54imbi2d 219 . . 3  |-  ( j  =  0  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ 0 ) )  =  ( ( * `  A
) ^ 0 ) ) ) )
6 oveq2 5520 . . . . . 6  |-  ( j  =  k  ->  ( A ^ j )  =  ( A ^ k
) )
76fveq2d 5182 . . . . 5  |-  ( j  =  k  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ k ) ) )
8 oveq2 5520 . . . . 5  |-  ( j  =  k  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
k ) )
97, 8eqeq12d 2054 . . . 4  |-  ( j  =  k  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) )
109imbi2d 219 . . 3  |-  ( j  =  k  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ k
) )  =  ( ( * `  A
) ^ k ) ) ) )
11 oveq2 5520 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  ( A ^ j )  =  ( A ^ (
k  +  1 ) ) )
1211fveq2d 5182 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ ( k  +  1 ) ) ) )
13 oveq2 5520 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
1412, 13eqeq12d 2054 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) )
1514imbi2d 219 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ (
k  +  1 ) ) )  =  ( ( * `  A
) ^ ( k  +  1 ) ) ) ) )
16 oveq2 5520 . . . . . 6  |-  ( j  =  N  ->  ( A ^ j )  =  ( A ^ N
) )
1716fveq2d 5182 . . . . 5  |-  ( j  =  N  ->  (
* `  ( A ^ j ) )  =  ( * `  ( A ^ N ) ) )
18 oveq2 5520 . . . . 5  |-  ( j  =  N  ->  (
( * `  A
) ^ j )  =  ( ( * `
 A ) ^ N ) )
1917, 18eqeq12d 2054 . . . 4  |-  ( j  =  N  ->  (
( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j )  <->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) )
2019imbi2d 219 . . 3  |-  ( j  =  N  ->  (
( A  e.  CC  ->  ( * `  ( A ^ j ) )  =  ( ( * `
 A ) ^
j ) )  <->  ( A  e.  CC  ->  ( * `  ( A ^ N
) )  =  ( ( * `  A
) ^ N ) ) ) )
21 exp0 9259 . . . . 5  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2221fveq2d 5182 . . . 4  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( * ` 
1 ) )
23 cjcl 9448 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
24 exp0 9259 . . . . . 6  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  1 )
25 1re 7026 . . . . . . 7  |-  1  e.  RR
26 cjre 9482 . . . . . . 7  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
2725, 26ax-mp 7 . . . . . 6  |-  ( * `
 1 )  =  1
2824, 27syl6eqr 2090 . . . . 5  |-  ( ( * `  A )  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
2923, 28syl 14 . . . 4  |-  ( A  e.  CC  ->  (
( * `  A
) ^ 0 )  =  ( * ` 
1 ) )
3022, 29eqtr4d 2075 . . 3  |-  ( A  e.  CC  ->  (
* `  ( A ^ 0 ) )  =  ( ( * `
 A ) ^
0 ) )
31 expp1 9262 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
3231fveq2d 5182 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( * `  ( ( A ^
k )  x.  A
) ) )
33 expcl 9273 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
34 simpl 102 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  ->  A  e.  CC )
35 cjmul 9485 . . . . . . . . . 10  |-  ( ( ( A ^ k
)  e.  CC  /\  A  e.  CC )  ->  ( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3633, 34, 35syl2anc 391 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( A ^ k
)  x.  A ) )  =  ( ( * `  ( A ^ k ) )  x.  ( * `  A ) ) )
3732, 36eqtrd 2072 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
3837adantr 261 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 ( A ^
k ) )  x.  ( * `  A
) ) )
39 oveq1 5519 . . . . . . . 8  |-  ( ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k )  ->  (
( * `  ( A ^ k ) )  x.  ( * `  A ) )  =  ( ( ( * `
 A ) ^
k )  x.  (
* `  A )
) )
40 expp1 9262 . . . . . . . . . 10  |-  ( ( ( * `  A
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4123, 40sylan 267 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  A ) ^ (
k  +  1 ) )  =  ( ( ( * `  A
) ^ k )  x.  ( * `  A ) ) )
4241eqcomd 2045 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( ( ( * `
 A ) ^
k )  x.  (
* `  A )
)  =  ( ( * `  A ) ^ ( k  +  1 ) ) )
4339, 42sylan9eqr 2094 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( ( * `  ( A ^ k ) )  x.  ( * `
 A ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4438, 43eqtrd 2072 . . . . . 6  |-  ( ( ( A  e.  CC  /\  k  e.  NN0 )  /\  ( * `  ( A ^ k ) )  =  ( ( * `
 A ) ^
k ) )  -> 
( * `  ( A ^ ( k  +  1 ) ) )  =  ( ( * `
 A ) ^
( k  +  1 ) ) )
4544exp31 346 . . . . 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 8352 . 2  |-  ( N  e.  NN0  ->  ( A  e.  CC  ->  (
* `  ( A ^ N ) )  =  ( ( * `  A ) ^ N
) ) )
4948impcom 116 1  |-  ( ( A  e.  CC  /\  N  e.  NN0 )  -> 
( * `  ( A ^ N ) )  =  ( ( * `
 A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   ` cfv 4902  (class class class)co 5512   CCcc 6887   RRcr 6888   0cc0 6889   1c1 6890    + caddc 6892    x. cmul 6894   NN0cn0 8181   ^cexp 9254   *ccj 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212  df-iexp 9255  df-cj 9442  df-re 9443  df-im 9444
This theorem is referenced by:  cjexpd  9558
  Copyright terms: Public domain W3C validator