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Theorem modqexp 10649
Description: Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.)
Hypotheses
Ref Expression
modqexp.a  |-  ( ph  ->  A  e.  ZZ )
modqexp.b  |-  ( ph  ->  B  e.  ZZ )
modqexp.c  |-  ( ph  ->  C  e.  NN0 )
modqexp.dq  |-  ( ph  ->  D  e.  QQ )
modqexp.dgt0  |-  ( ph  ->  0  <  D )
modqexp.mod  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
Assertion
Ref Expression
modqexp  |-  ( ph  ->  ( ( A ^ C )  mod  D
)  =  ( ( B ^ C )  mod  D ) )

Proof of Theorem modqexp
Dummy variables  w  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 modqexp.c . 2  |-  ( ph  ->  C  e.  NN0 )
2 oveq2 5885 . . . . . 6  |-  ( w  =  0  ->  ( A ^ w )  =  ( A ^ 0 ) )
32oveq1d 5892 . . . . 5  |-  ( w  =  0  ->  (
( A ^ w
)  mod  D )  =  ( ( A ^ 0 )  mod 
D ) )
4 oveq2 5885 . . . . . 6  |-  ( w  =  0  ->  ( B ^ w )  =  ( B ^ 0 ) )
54oveq1d 5892 . . . . 5  |-  ( w  =  0  ->  (
( B ^ w
)  mod  D )  =  ( ( B ^ 0 )  mod 
D ) )
63, 5eqeq12d 2192 . . . 4  |-  ( w  =  0  ->  (
( ( A ^
w )  mod  D
)  =  ( ( B ^ w )  mod  D )  <->  ( ( A ^ 0 )  mod 
D )  =  ( ( B ^ 0 )  mod  D ) ) )
76imbi2d 230 . . 3  |-  ( w  =  0  ->  (
( ph  ->  ( ( A ^ w )  mod  D )  =  ( ( B ^
w )  mod  D
) )  <->  ( ph  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) ) ) )
8 oveq2 5885 . . . . . 6  |-  ( w  =  k  ->  ( A ^ w )  =  ( A ^ k
) )
98oveq1d 5892 . . . . 5  |-  ( w  =  k  ->  (
( A ^ w
)  mod  D )  =  ( ( A ^ k )  mod 
D ) )
10 oveq2 5885 . . . . . 6  |-  ( w  =  k  ->  ( B ^ w )  =  ( B ^ k
) )
1110oveq1d 5892 . . . . 5  |-  ( w  =  k  ->  (
( B ^ w
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )
129, 11eqeq12d 2192 . . . 4  |-  ( w  =  k  ->  (
( ( A ^
w )  mod  D
)  =  ( ( B ^ w )  mod  D )  <->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
) )
1312imbi2d 230 . . 3  |-  ( w  =  k  ->  (
( ph  ->  ( ( A ^ w )  mod  D )  =  ( ( B ^
w )  mod  D
) )  <->  ( ph  ->  ( ( A ^
k )  mod  D
)  =  ( ( B ^ k )  mod  D ) ) ) )
14 oveq2 5885 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( A ^ w )  =  ( A ^ (
k  +  1 ) ) )
1514oveq1d 5892 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( A ^ w
)  mod  D )  =  ( ( A ^ ( k  +  1 ) )  mod 
D ) )
16 oveq2 5885 . . . . . 6  |-  ( w  =  ( k  +  1 )  ->  ( B ^ w )  =  ( B ^ (
k  +  1 ) ) )
1716oveq1d 5892 . . . . 5  |-  ( w  =  ( k  +  1 )  ->  (
( B ^ w
)  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) )
1815, 17eqeq12d 2192 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
( ( A ^
w )  mod  D
)  =  ( ( B ^ w )  mod  D )  <->  ( ( A ^ ( k  +  1 ) )  mod 
D )  =  ( ( B ^ (
k  +  1 ) )  mod  D ) ) )
1918imbi2d 230 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
( ph  ->  ( ( A ^ w )  mod  D )  =  ( ( B ^
w )  mod  D
) )  <->  ( ph  ->  ( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( B ^ ( k  +  1 ) )  mod  D ) ) ) )
20 oveq2 5885 . . . . . 6  |-  ( w  =  C  ->  ( A ^ w )  =  ( A ^ C
) )
2120oveq1d 5892 . . . . 5  |-  ( w  =  C  ->  (
( A ^ w
)  mod  D )  =  ( ( A ^ C )  mod 
D ) )
22 oveq2 5885 . . . . . 6  |-  ( w  =  C  ->  ( B ^ w )  =  ( B ^ C
) )
2322oveq1d 5892 . . . . 5  |-  ( w  =  C  ->  (
( B ^ w
)  mod  D )  =  ( ( B ^ C )  mod 
D ) )
2421, 23eqeq12d 2192 . . . 4  |-  ( w  =  C  ->  (
( ( A ^
w )  mod  D
)  =  ( ( B ^ w )  mod  D )  <->  ( ( A ^ C )  mod 
D )  =  ( ( B ^ C
)  mod  D )
) )
2524imbi2d 230 . . 3  |-  ( w  =  C  ->  (
( ph  ->  ( ( A ^ w )  mod  D )  =  ( ( B ^
w )  mod  D
) )  <->  ( ph  ->  ( ( A ^ C )  mod  D
)  =  ( ( B ^ C )  mod  D ) ) ) )
26 modqexp.a . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
2726zcnd 9378 . . . . . 6  |-  ( ph  ->  A  e.  CC )
28 exp0 10526 . . . . . 6  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2927, 28syl 14 . . . . 5  |-  ( ph  ->  ( A ^ 0 )  =  1 )
30 modqexp.b . . . . . . 7  |-  ( ph  ->  B  e.  ZZ )
3130zcnd 9378 . . . . . 6  |-  ( ph  ->  B  e.  CC )
32 exp0 10526 . . . . . 6  |-  ( B  e.  CC  ->  ( B ^ 0 )  =  1 )
3331, 32syl 14 . . . . 5  |-  ( ph  ->  ( B ^ 0 )  =  1 )
3429, 33eqtr4d 2213 . . . 4  |-  ( ph  ->  ( A ^ 0 )  =  ( B ^ 0 ) )
3534oveq1d 5892 . . 3  |-  ( ph  ->  ( ( A ^
0 )  mod  D
)  =  ( ( B ^ 0 )  mod  D ) )
36 zexpcl 10537 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
3726, 36sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  ZZ )
3837adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A ^ k
)  e.  ZZ )
39 zexpcl 10537 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  ZZ )
4030, 39sylan 283 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( B ^ k )  e.  ZZ )
4140adantr 276 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( B ^ k
)  e.  ZZ )
4226ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  A  e.  ZZ )
4330ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  B  e.  ZZ )
44 modqexp.dq . . . . . . . . 9  |-  ( ph  ->  D  e.  QQ )
4544ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  D  e.  QQ )
46 modqexp.dgt0 . . . . . . . . 9  |-  ( ph  ->  0  <  D )
4746ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
0  <  D )
48 simpr 110 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
k )  mod  D
)  =  ( ( B ^ k )  mod  D ) )
49 modqexp.mod . . . . . . . . 9  |-  ( ph  ->  ( A  mod  D
)  =  ( B  mod  D ) )
5049ad2antrr 488 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A  mod  D
)  =  ( B  mod  D ) )
5138, 41, 42, 43, 45, 47, 48, 50modqmul12d 10380 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( ( A ^ k )  x.  A )  mod  D
)  =  ( ( ( B ^ k
)  x.  B )  mod  D ) )
5227ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  A  e.  CC )
53 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
5453adantr 276 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
k  e.  NN0 )
55 expp1 10529 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5652, 54, 55syl2anc 411 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
5756oveq1d 5892 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( ( A ^ k
)  x.  A )  mod  D ) )
5831ad2antrr 488 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  ->  B  e.  CC )
59 expp1 10529 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  k  e.  NN0 )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
6058, 54, 59syl2anc 411 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( B ^ (
k  +  1 ) )  =  ( ( B ^ k )  x.  B ) )
6160oveq1d 5892 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( B ^
( k  +  1 ) )  mod  D
)  =  ( ( ( B ^ k
)  x.  B )  mod  D ) )
6251, 57, 613eqtr4d 2220 . . . . . 6  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D ) )  -> 
( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( B ^ ( k  +  1 ) )  mod  D ) )
6362ex 115 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
( A ^ k
)  mod  D )  =  ( ( B ^ k )  mod 
D )  ->  (
( A ^ (
k  +  1 ) )  mod  D )  =  ( ( B ^ ( k  +  1 ) )  mod 
D ) ) )
6463expcom 116 . . . 4  |-  ( k  e.  NN0  ->  ( ph  ->  ( ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )  ->  ( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( B ^ ( k  +  1 ) )  mod  D ) ) ) )
6564a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( (
ph  ->  ( ( A ^ k )  mod 
D )  =  ( ( B ^ k
)  mod  D )
)  ->  ( ph  ->  ( ( A ^
( k  +  1 ) )  mod  D
)  =  ( ( B ^ ( k  +  1 ) )  mod  D ) ) ) )
667, 13, 19, 25, 35, 65nn0ind 9369 . 2  |-  ( C  e.  NN0  ->  ( ph  ->  ( ( A ^ C )  mod  D
)  =  ( ( B ^ C )  mod  D ) ) )
671, 66mpcom 36 1  |-  ( ph  ->  ( ( A ^ C )  mod  D
)  =  ( ( B ^ C )  mod  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814    + caddc 7816    x. cmul 7818    < clt 7994   NN0cn0 9178   ZZcz 9255   QQcq 9621    mod cmo 10324   ^cexp 10521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522
This theorem is referenced by:  dvdsmodexp  11804  odzdvds  12247  lgsmod  14466  lgsne0  14478
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