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Theorem modexp 12256
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 1017 . 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 1217 . 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 6283 . . . . . 6  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
54oveq1d 6290 . . . . 5  |-  ( x  =  0  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ 0 )  mod 
D ) )
6 oveq2 6283 . . . . . 6  |-  ( x  =  0  ->  ( B ^ x )  =  ( B ^ 0 ) )
76oveq1d 6290 . . . . 5  |-  ( x  =  0  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ 0 )  mod 
D ) )
85, 7eqeq12d 2482 . . . 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 6283 . . . . . 6  |-  ( x  =  k  ->  ( A ^ x )  =  ( A ^ k
) )
1110oveq1d 6290 . . . . 5  |-  ( x  =  k  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ k )  mod 
D ) )
12 oveq2 6283 . . . . . 6  |-  ( x  =  k  ->  ( B ^ x )  =  ( B ^ k
) )
1312oveq1d 6290 . . . . 5  |-  ( x  =  k  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )
1411, 13eqeq12d 2482 . . . 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 6283 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( A ^ x )  =  ( A ^ (
k  +  1 ) ) )
1716oveq1d 6290 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ ( k  +  1 ) )  mod 
D ) )
18 oveq2 6283 . . . . . 6  |-  ( x  =  ( k  +  1 )  ->  ( B ^ x )  =  ( B ^ (
k  +  1 ) ) )
1918oveq1d 6290 . . . . 5  |-  ( x  =  ( k  +  1 )  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) )
2017, 19eqeq12d 2482 . . . 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 6283 . . . . . 6  |-  ( x  =  C  ->  ( A ^ x )  =  ( A ^ C
) )
2322oveq1d 6290 . . . . 5  |-  ( x  =  C  ->  (
( A ^ x
)  mod  D )  =  ( ( A ^ C )  mod 
D ) )
24 oveq2 6283 . . . . . 6  |-  ( x  =  C  ->  ( B ^ x )  =  ( B ^ C
) )
2524oveq1d 6290 . . . . 5  |-  ( x  =  C  ->  (
( B ^ x
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
2623, 25eqeq12d 2482 . . . 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 10858 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
29 exp0 12126 . . . . . . 7  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
3028, 29syl 16 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A ^ 0 )  =  1 )
31 zcn 10858 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
32 exp0 12126 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3331, 32syl 16 . . . . . . 7  |-  ( B  e.  ZZ  ->  ( B ^ 0 )  =  1 )
3433eqcomd 2468 . . . . . 6  |-  ( B  e.  ZZ  ->  1  =  ( B ^
0 ) )
3530, 34sylan9eq 2521 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ^ 0 )  =  ( B ^ 0 ) )
3635oveq1d 6290 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) )
37363ad2ant1 1012 . . 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 1108 . . . . . . . 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 991 . . . . . . . 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 12137 . . . . . . . 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 1109 . . . . . . . 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 12137 . . . . . . . 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 1025 . . . . . . 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 993 . . . . . . 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 1026 . . . . . . 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 11997 . . . . . 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 10956 . . . . . . . 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 12129 . . . . . . . 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 6290 . . . . . 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 10956 . . . . . . . 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 12129 . . . . . . . 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 6290 . . . . . 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 2511 . . . . 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 1190 . . . 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 10946 . 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 968    = wceq 1374    e. wcel 1762  (class class class)co 6275   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   NN0cn0 10784   ZZcz 10853   RR+crp 11209    mod cmo 11952   ^cexp 12122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123
This theorem is referenced by:  fermltl  14162  odzdvds  14170  lgslem4  23295  lgsmod  23317  lgsne0  23329
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