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Theorem modexp 11252
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 981 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  C  e.  NN0 )
2 id 19 . . 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 1176 . 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 5882 . . . . . 6  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
54oveq1d 5889 . . . . 5  |-  ( x  =  0  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ 0 )  mod 
D ) )
6 oveq2 5882 . . . . . 6  |-  ( x  =  0  ->  ( B ^ x )  =  ( B ^ 0 ) )
76oveq1d 5889 . . . . 5  |-  ( x  =  0  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ 0 )  mod 
D ) )
85, 7eqeq12d 2310 . . . 4  |-  ( x  =  0  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) ) )
98imbi2d 307 . . 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 5882 . . . . . 6  |-  ( x  =  k  ->  ( A ^ x )  =  ( A ^ k
) )
1110oveq1d 5889 . . . . 5  |-  ( x  =  k  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ k )  mod 
D ) )
12 oveq2 5882 . . . . . 6  |-  ( x  =  k  ->  ( B ^ x )  =  ( B ^ k
) )
1312oveq1d 5889 . . . . 5  |-  ( x  =  k  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )
1411, 13eqeq12d 2310 . . . 4  |-  ( x  =  k  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
) )
1514imbi2d 307 . . 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 5882 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A ^ x )  =  ( A ^ (
k  +  1 ) ) )
1716oveq1d 5889 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ ( k  +  1 ) )  mod 
D ) )
18 oveq2 5882 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( B ^ x )  =  ( B ^ (
k  +  1 ) ) )
1918oveq1d 5889 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) )
2017, 19eqeq12d 2310 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) )
2120imbi2d 307 . . 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 5882 . . . . . 6  |-  ( x  =  C  ->  ( A ^ x )  =  ( A ^ C
) )
2322oveq1d 5889 . . . . 5  |-  ( x  =  C  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ C )  mod 
D ) )
24 oveq2 5882 . . . . . 6  |-  ( x  =  C  ->  ( B ^ x )  =  ( B ^ C
) )
2524oveq1d 5889 . . . . 5  |-  ( x  =  C  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
2623, 25eqeq12d 2310 . . . 4  |-  ( x  =  C  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) )
2726imbi2d 307 . . 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 10045 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
29 exp0 11124 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3028, 29syl 15 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A ^ 0 )  =  1 )
31 zcn 10045 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
32 exp0 11124 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3331, 32syl 15 . . . . . . 7  |-  ( B  e.  ZZ  ->  ( B ^ 0 )  =  1 )
3433eqcomd 2301 . . . . . 6  |-  ( B  e.  ZZ  ->  1  =  ( B ^
0 ) )
3530, 34sylan9eq 2348 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ^ 0 )  =  ( B ^ 0 ) )
3635oveq1d 5889 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) )
37363ad2ant1 976 . . 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 1072 . . . . . . . 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 955 . . . . . . . 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 11134 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
4138, 39, 40syl2anc 642 . . . . . . 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 1073 . . . . . . . 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 11134 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  ZZ )
4442, 39, 43syl2anc 642 . . . . . . 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 989 . . . . . . 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 957 . . . . . . 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 990 . . . . . . 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 11019 . . . . . 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 10134 . . . . . . . 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 11126 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5149, 39, 50syl2anc 642 . . . . . . 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 5889 . . . . . 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 10134 . . . . . . . 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 11126 . . . . . . . 8  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
5553, 39, 54syl2anc 642 . . . . . . 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 5889 . . . . . 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 2338 . . . . 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 1150 . . . 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 23 . . 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 10124 . 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 56 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NN0cn0 9981   ZZcz 10040   RR+crp 10370    mod cmo 10989   ^cexp 11120
This theorem is referenced by:  fermltl  12868  odzdvds  12876  lgslem4  20554  lgsmod  20576  lgsne0  20588
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-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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