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Theorem iexpcyc 11223
Description: Taking  _i to the  K-th power is the same as using the  K  mod  4 -th power instead, by i4 11221. (Contributed by Mario Carneiro, 7-Jul-2014.)
Assertion
Ref Expression
iexpcyc  |-  ( K  e.  ZZ  ->  (
_i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )

Proof of Theorem iexpcyc
StepHypRef Expression
1 zre 10044 . . . 4  |-  ( K  e.  ZZ  ->  K  e.  RR )
2 4re 9835 . . . . 5  |-  4  e.  RR
3 4pos 9848 . . . . 5  |-  0  <  4
42, 3elrpii 10373 . . . 4  |-  4  e.  RR+
5 modval 10991 . . . 4  |-  ( ( K  e.  RR  /\  4  e.  RR+ )  -> 
( K  mod  4
)  =  ( K  -  ( 4  x.  ( |_ `  ( K  /  4 ) ) ) ) )
61, 4, 5sylancl 643 . . 3  |-  ( K  e.  ZZ  ->  ( K  mod  4 )  =  ( K  -  (
4  x.  ( |_
`  ( K  / 
4 ) ) ) ) )
76oveq2d 5890 . 2  |-  ( K  e.  ZZ  ->  (
_i ^ ( K  mod  4 ) )  =  ( _i ^
( K  -  (
4  x.  ( |_
`  ( K  / 
4 ) ) ) ) ) )
8 4nn 9895 . . . . . 6  |-  4  e.  NN
98nnzi 10063 . . . . 5  |-  4  e.  ZZ
10 nndivre 9797 . . . . . . 7  |-  ( ( K  e.  RR  /\  4  e.  NN )  ->  ( K  /  4
)  e.  RR )
111, 8, 10sylancl 643 . . . . . 6  |-  ( K  e.  ZZ  ->  ( K  /  4 )  e.  RR )
1211flcld 10946 . . . . 5  |-  ( K  e.  ZZ  ->  ( |_ `  ( K  / 
4 ) )  e.  ZZ )
13 zmulcl 10082 . . . . 5  |-  ( ( 4  e.  ZZ  /\  ( |_ `  ( K  /  4 ) )  e.  ZZ )  -> 
( 4  x.  ( |_ `  ( K  / 
4 ) ) )  e.  ZZ )
149, 12, 13sylancr 644 . . . 4  |-  ( K  e.  ZZ  ->  (
4  x.  ( |_
`  ( K  / 
4 ) ) )  e.  ZZ )
15 ax-icn 8812 . . . . 5  |-  _i  e.  CC
16 ine0 9231 . . . . 5  |-  _i  =/=  0
17 expsub 11165 . . . . 5  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( K  e.  ZZ  /\  ( 4  x.  ( |_ `  ( K  /  4
) ) )  e.  ZZ ) )  -> 
( _i ^ ( K  -  ( 4  x.  ( |_ `  ( K  /  4
) ) ) ) )  =  ( ( _i ^ K )  /  ( _i ^
( 4  x.  ( |_ `  ( K  / 
4 ) ) ) ) ) )
1815, 16, 17mpanl12 663 . . . 4  |-  ( ( K  e.  ZZ  /\  ( 4  x.  ( |_ `  ( K  / 
4 ) ) )  e.  ZZ )  -> 
( _i ^ ( K  -  ( 4  x.  ( |_ `  ( K  /  4
) ) ) ) )  =  ( ( _i ^ K )  /  ( _i ^
( 4  x.  ( |_ `  ( K  / 
4 ) ) ) ) ) )
1914, 18mpdan 649 . . 3  |-  ( K  e.  ZZ  ->  (
_i ^ ( K  -  ( 4  x.  ( |_ `  ( K  /  4 ) ) ) ) )  =  ( ( _i ^ K )  /  (
_i ^ ( 4  x.  ( |_ `  ( K  /  4
) ) ) ) ) )
20 expmulz 11164 . . . . . . . 8  |-  ( ( ( _i  e.  CC  /\  _i  =/=  0 )  /\  ( 4  e.  ZZ  /\  ( |_
`  ( K  / 
4 ) )  e.  ZZ ) )  -> 
( _i ^ (
4  x.  ( |_
`  ( K  / 
4 ) ) ) )  =  ( ( _i ^ 4 ) ^ ( |_ `  ( K  /  4
) ) ) )
2115, 16, 20mpanl12 663 . . . . . . 7  |-  ( ( 4  e.  ZZ  /\  ( |_ `  ( K  /  4 ) )  e.  ZZ )  -> 
( _i ^ (
4  x.  ( |_
`  ( K  / 
4 ) ) ) )  =  ( ( _i ^ 4 ) ^ ( |_ `  ( K  /  4
) ) ) )
229, 12, 21sylancr 644 . . . . . 6  |-  ( K  e.  ZZ  ->  (
_i ^ ( 4  x.  ( |_ `  ( K  /  4
) ) ) )  =  ( ( _i
^ 4 ) ^
( |_ `  ( K  /  4 ) ) ) )
23 i4 11221 . . . . . . . 8  |-  ( _i
^ 4 )  =  1
2423oveq1i 5884 . . . . . . 7  |-  ( ( _i ^ 4 ) ^ ( |_ `  ( K  /  4
) ) )  =  ( 1 ^ ( |_ `  ( K  / 
4 ) ) )
25 1exp 11147 . . . . . . . 8  |-  ( ( |_ `  ( K  /  4 ) )  e.  ZZ  ->  (
1 ^ ( |_
`  ( K  / 
4 ) ) )  =  1 )
2612, 25syl 15 . . . . . . 7  |-  ( K  e.  ZZ  ->  (
1 ^ ( |_
`  ( K  / 
4 ) ) )  =  1 )
2724, 26syl5eq 2340 . . . . . 6  |-  ( K  e.  ZZ  ->  (
( _i ^ 4 ) ^ ( |_
`  ( K  / 
4 ) ) )  =  1 )
2822, 27eqtrd 2328 . . . . 5  |-  ( K  e.  ZZ  ->  (
_i ^ ( 4  x.  ( |_ `  ( K  /  4
) ) ) )  =  1 )
2928oveq2d 5890 . . . 4  |-  ( K  e.  ZZ  ->  (
( _i ^ K
)  /  ( _i
^ ( 4  x.  ( |_ `  ( K  /  4 ) ) ) ) )  =  ( ( _i ^ K )  /  1
) )
30 expclz 11144 . . . . . 6  |-  ( ( _i  e.  CC  /\  _i  =/=  0  /\  K  e.  ZZ )  ->  (
_i ^ K )  e.  CC )
3115, 16, 30mp3an12 1267 . . . . 5  |-  ( K  e.  ZZ  ->  (
_i ^ K )  e.  CC )
3231div1d 9544 . . . 4  |-  ( K  e.  ZZ  ->  (
( _i ^ K
)  /  1 )  =  ( _i ^ K ) )
3329, 32eqtrd 2328 . . 3  |-  ( K  e.  ZZ  ->  (
( _i ^ K
)  /  ( _i
^ ( 4  x.  ( |_ `  ( K  /  4 ) ) ) ) )  =  ( _i ^ K
) )
3419, 33eqtrd 2328 . 2  |-  ( K  e.  ZZ  ->  (
_i ^ ( K  -  ( 4  x.  ( |_ `  ( K  /  4 ) ) ) ) )  =  ( _i ^ K
) )
357, 34eqtrd 2328 1  |-  ( K  e.  ZZ  ->  (
_i ^ ( K  mod  4 ) )  =  ( _i ^ K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   4c4 9813   ZZcz 10040   RR+crp 10370   |_cfl 10940    mod cmo 10989   ^cexp 11120
This theorem is referenced by:  iblitg  19139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121
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