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Theorem modexp 12407
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 1031 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  RR+ )  /\  ( A  mod  D )  =  ( B  mod  D
) )  ->  C  e.  NN0 )
2 id 23 . . 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 1258 . 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 6310 . . . . . 6  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
54oveq1d 6317 . . . . 5  |-  ( x  =  0  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ 0 )  mod 
D ) )
6 oveq2 6310 . . . . . 6  |-  ( x  =  0  ->  ( B ^ x )  =  ( B ^ 0 ) )
76oveq1d 6317 . . . . 5  |-  ( x  =  0  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ 0 )  mod 
D ) )
85, 7eqeq12d 2444 . . . 4  |-  ( x  =  0  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) ) )
98imbi2d 317 . . 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 6310 . . . . . 6  |-  ( x  =  k  ->  ( A ^ x )  =  ( A ^ k
) )
1110oveq1d 6317 . . . . 5  |-  ( x  =  k  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ k )  mod 
D ) )
12 oveq2 6310 . . . . . 6  |-  ( x  =  k  ->  ( B ^ x )  =  ( B ^ k
) )
1312oveq1d 6317 . . . . 5  |-  ( x  =  k  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )
1411, 13eqeq12d 2444 . . . 4  |-  ( x  =  k  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
) )
1514imbi2d 317 . . 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 6310 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A ^ x )  =  ( A ^ (
k  +  1 ) ) )
1716oveq1d 6317 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ ( k  +  1 ) )  mod 
D ) )
18 oveq2 6310 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( B ^ x )  =  ( B ^ (
k  +  1 ) ) )
1918oveq1d 6317 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) )
2017, 19eqeq12d 2444 . . . 4  |-  ( x  =  ( k  +  1 )  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) )
2120imbi2d 317 . . 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 6310 . . . . . 6  |-  ( x  =  C  ->  ( A ^ x )  =  ( A ^ C
) )
2322oveq1d 6317 . . . . 5  |-  ( x  =  C  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ C )  mod 
D ) )
24 oveq2 6310 . . . . . 6  |-  ( x  =  C  ->  ( B ^ x )  =  ( B ^ C
) )
2524oveq1d 6317 . . . . 5  |-  ( x  =  C  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
2623, 25eqeq12d 2444 . . . 4  |-  ( x  =  C  ->  (
( ( A ^
x )  mod  D
)  =  ( ( B ^ x )  mod  D )  <->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) )
2726imbi2d 317 . . 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 10943 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
29 exp0 12276 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3028, 29syl 17 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A ^ 0 )  =  1 )
31 zcn 10943 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
32 exp0 12276 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3331, 32syl 17 . . . . . . 7  |-  ( B  e.  ZZ  ->  ( B ^ 0 )  =  1 )
3433eqcomd 2430 . . . . . 6  |-  ( B  e.  ZZ  ->  1  =  ( B ^
0 ) )
3530, 34sylan9eq 2483 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ^ 0 )  =  ( B ^ 0 ) )
3635oveq1d 6317 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) )
37363ad2ant1 1026 . . 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 1122 . . . . . . . 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 1005 . . . . . . . 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 12287 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
4138, 39, 40syl2anc 665 . . . . . . 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 1123 . . . . . . . 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 12287 . . . . . . . 8  |-  ( ( B  e.  ZZ  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  ZZ )
4442, 39, 43syl2anc 665 . . . . . . 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 1039 . . . . . . 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 1007 . . . . . . 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 1040 . . . . . . 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 12144 . . . . . 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 11042 . . . . . . . 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 12279 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5149, 39, 50syl2anc 665 . . . . . . 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 6317 . . . . . 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 11042 . . . . . . . 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 12279 . . . . . . . 8  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
5553, 39, 54syl2anc 665 . . . . . . 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 6317 . . . . . 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 2473 . . . . 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 1204 . . . 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 29 . . 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 11031 . 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 62 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868  (class class class)co 6302   CCcc 9538   0cc0 9540   1c1 9541    + caddc 9543    x. cmul 9545   NN0cn0 10870   ZZcz 10938   RR+crp 11303    mod cmo 12096   ^cexp 12272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7959  df-inf 7960  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-n0 10871  df-z 10939  df-uz 11161  df-rp 11304  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273
This theorem is referenced by:  fermltl  14720  odzdvds  14728  odzdvdsOLD  14734  lgslem4  24214  lgsmod  24236  lgsne0  24248  41prothprmlem2  38630
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