MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  modexp Structured version   Unicode version

Theorem modexp 12282
Description: Exponentiation property of the modulo operation. (Contributed by Mario Carneiro, 28-Feb-2014.)
Assertion
Ref Expression
modexp  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ C
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )

Proof of Theorem modexp
Dummy variables  x  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2l 1023 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  C  e.  NN0 )
2 id 22 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) ) )
323adant2l 1223 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) ) )
4 oveq2 6289 . . . . . 6  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
54oveq1d 6296 . . . . 5  |-  ( x  =  0  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ 0 )  mod 
D ) )
6 oveq2 6289 . . . . . 6  |-  ( x  =  0  ->  ( B ^ x )  =  ( B ^ 0 ) )
76oveq1d 6296 . . . . 5  |-  ( x  =  0  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ 0 )  mod 
D ) )
85, 7eqeq12d 2465 . . . 4  |-  ( x  =  0  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) ) )
98imbi2d 316 . . 3  |-  ( x  =  0  ->  (
( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ x
)  mod  D )  =  ( ( B ^ x )  mod 
D ) )  <->  ( (
( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) ) ) )
10 oveq2 6289 . . . . . 6  |-  ( x  =  k  ->  ( A ^ x )  =  ( A ^ k
) )
1110oveq1d 6296 . . . . 5  |-  ( x  =  k  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ k )  mod 
D ) )
12 oveq2 6289 . . . . . 6  |-  ( x  =  k  ->  ( B ^ x )  =  ( B ^ k
) )
1312oveq1d 6296 . . . . 5  |-  ( x  =  k  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )
1411, 13eqeq12d 2465 . . . 4  |-  ( x  =  k  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
) )
1514imbi2d 316 . . 3  |-  ( x  =  k  ->  (
( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ x
)  mod  D )  =  ( ( B ^ x )  mod 
D ) )  <->  ( (
( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
) ) )
16 oveq2 6289 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A ^ x )  =  ( A ^ (
k  +  1 ) ) )
1716oveq1d 6296 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ ( k  +  1 ) )  mod 
D ) )
18 oveq2 6289 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( B ^ x )  =  ( B ^ (
k  +  1 ) ) )
1918oveq1d 6296 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) )
2017, 19eqeq12d 2465 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) )
2120imbi2d 316 . . 3  |-  ( x  =  ( k  +  1 )  ->  (
( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ x
)  mod  D )  =  ( ( B ^ x )  mod 
D ) )  <->  ( (
( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) ) )
22 oveq2 6289 . . . . . 6  |-  ( x  =  C  ->  ( A ^ x )  =  ( A ^ C
) )
2322oveq1d 6296 . . . . 5  |-  ( x  =  C  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ C )  mod 
D ) )
24 oveq2 6289 . . . . . 6  |-  ( x  =  C  ->  ( B ^ x )  =  ( B ^ C
) )
2524oveq1d 6296 . . . . 5  |-  ( x  =  C  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
2623, 25eqeq12d 2465 . . . 4  |-  ( x  =  C  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) )
2726imbi2d 316 . . 3  |-  ( x  =  C  ->  (
( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ x
)  mod  D )  =  ( ( B ^ x )  mod 
D ) )  <->  ( (
( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) ) )
28 zcn 10876 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
29 exp0 12151 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3028, 29syl 16 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A ^ 0 )  =  1 )
31 zcn 10876 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
32 exp0 12151 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3331, 32syl 16 . . . . . . 7  |-  ( B  e.  ZZ  ->  ( B ^ 0 )  =  1 )
3433eqcomd 2451 . . . . . 6  |-  ( B  e.  ZZ  ->  1  =  ( B ^
0 ) )
3530, 34sylan9eq 2504 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ^ 0 )  =  ( B ^ 0 ) )
3635oveq1d 6296 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) )
37363ad2ant1 1018 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) )
38 simp21l 1114 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  A  e.  ZZ )
39 simp1 997 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
k  e.  NN0 )
40 zexpcl 12162 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
4138, 39, 40syl2anc 661 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A ^ k
)  e.  ZZ )
42 simp21r 1115 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  B  e.  ZZ )
43 zexpcl 12162 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  ZZ )
4442, 39, 43syl2anc 661 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( B ^ k
)  e.  ZZ )
45 simp22 1031 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  D  e.  RR+ )
46 simp3 999 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
k )  mod  D
)  =  ( ( B ^ k )  mod  D ) )
47 simp23 1032 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A  mod  D
)  =  ( B  mod  D ) )
4841, 44, 38, 42, 45, 46, 47modmul12d 12022 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( ( A ^ k )  x.  A )  mod  D
)  =  ( ( ( B ^ k
)  x.  B )  mod  D ) )
4938zcnd 10976 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  A  e.  CC )
50 expp1 12154 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5149, 39, 50syl2anc 661 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5251oveq1d 6296 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( ( A ^ k
)  x.  A )  mod  D ) )
5342zcnd 10976 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  B  e.  CC )
54 expp1 12154 . . . . . . . 8  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
5553, 39, 54syl2anc 661 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
5655oveq1d 6296 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( B ^
( k  +  1 ) )  mod  D
)  =  ( ( ( B ^ k
)  x.  B )  mod  D ) )
5748, 52, 563eqtr4d 2494 . . . . 5  |-  ( ( k  e.  NN0  /\  ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( B ^ ( k  +  1 ) )  mod  D ) )
58573exp 1196 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( ( A ^ k )  mod  D )  =  ( ( B ^
k )  mod  D
)  ->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) ) )
5958a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ (
k  +  1 ) )  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) ) ) )
609, 15, 21, 27, 37, 59nn0ind 10966 . 2  |-  ( C  e.  NN0  ->  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  D  e.  RR+  /\  ( A  mod  D
)  =  ( B  mod  D ) )  ->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) )
611, 3, 60sylc 60 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  (
( A ^ C
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   NN0cn0 10802   ZZcz 10871   RR+crp 11230    mod cmo 11977   ^cexp 12147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-fl 11910  df-mod 11978  df-seq 12089  df-exp 12148
This theorem is referenced by:  fermltl  14295  odzdvds  14303  lgslem4  23550  lgsmod  23572  lgsne0  23584
  Copyright terms: Public domain W3C validator