ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prmdvdsexp Unicode version

Theorem prmdvdsexp 11405
Description: A prime divides a positive power of an integer iff it divides the integer. (Contributed by Mario Carneiro, 24-Feb-2014.) (Revised by Mario Carneiro, 17-Jul-2014.)
Assertion
Ref Expression
prmdvdsexp  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A
) )

Proof of Theorem prmdvdsexp
Dummy variables  m  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5660 . . . . . . 7  |-  ( m  =  1  ->  ( A ^ m )  =  ( A ^ 1 ) )
21breq2d 3857 . . . . . 6  |-  ( m  =  1  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ 1 ) ) )
32bibi1d 231 . . . . 5  |-  ( m  =  1  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ 1 )  <-> 
P  ||  A )
) )
43imbi2d 228 . . . 4  |-  ( m  =  1  ->  (
( ( P  e. 
Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ m )  <-> 
P  ||  A )
)  <->  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) ) ) )
5 oveq2 5660 . . . . . . 7  |-  ( m  =  k  ->  ( A ^ m )  =  ( A ^ k
) )
65breq2d 3857 . . . . . 6  |-  ( m  =  k  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ k ) ) )
76bibi1d 231 . . . . 5  |-  ( m  =  k  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
) )
87imbi2d 228 . . . 4  |-  ( m  =  k  ->  (
( ( P  e. 
Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ m )  <-> 
P  ||  A )
)  <->  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
k )  <->  P  ||  A
) ) ) )
9 oveq2 5660 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  ( A ^ m )  =  ( A ^ (
k  +  1 ) ) )
109breq2d 3857 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ ( k  +  1 ) ) ) )
1110bibi1d 231 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  A )
) )
1211imbi2d 228 . . . 4  |-  ( m  =  ( k  +  1 )  ->  (
( ( P  e. 
Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ m )  <-> 
P  ||  A )
)  <->  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
( k  +  1 ) )  <->  P  ||  A
) ) ) )
13 oveq2 5660 . . . . . . 7  |-  ( m  =  N  ->  ( A ^ m )  =  ( A ^ N
) )
1413breq2d 3857 . . . . . 6  |-  ( m  =  N  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ N ) ) )
1514bibi1d 231 . . . . 5  |-  ( m  =  N  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
1615imbi2d 228 . . . 4  |-  ( m  =  N  ->  (
( ( P  e. 
Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ m )  <-> 
P  ||  A )
)  <->  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A
) ) ) )
17 zcn 8755 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1817adantl 271 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  CC )
1918exp1d 10081 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ 1 )  =  A )
2019breq2d 3857 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) )
21 nnnn0 8680 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
22 expp1 9962 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2318, 21, 22syl2an 283 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2423breq2d 3857 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  ( ( A ^ k )  x.  A ) ) )
25 simpll 496 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  P  e.  Prime )
26 simpr 108 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
27 zexpcl 9970 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
2826, 21, 27syl2an 283 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
k )  e.  ZZ )
29 simplr 497 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  A  e.  ZZ )
30 euclemma 11403 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A ^ k )  e.  ZZ  /\  A  e.  ZZ )  ->  ( P  ||  ( ( A ^ k )  x.  A )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) ) )
3125, 28, 29, 30syl3anc 1174 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( ( A ^
k )  x.  A
)  <->  ( P  ||  ( A ^ k )  \/  P  ||  A
) ) )
3224, 31bitrd 186 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
( P  ||  ( A ^ k )  \/  P  ||  A ) ) )
33 orbi1 741 . . . . . . . . 9  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  ( P  ||  A  \/  P  ||  A ) ) )
34 oridm 709 . . . . . . . . 9  |-  ( ( P  ||  A  \/  P  ||  A )  <->  P  ||  A
)
3533, 34syl6bb 194 . . . . . . . 8  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  P  ||  A
) )
3635bibi2d 230 . . . . . . 7  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
( k  +  1 ) )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) )  <->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3732, 36syl5ibcom 153 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3837expcom 114 . . . . 5  |-  ( k  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) ) )
3938a2d 26 . . . 4  |-  ( k  e.  NN  ->  (
( ( P  e. 
Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
)  ->  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
( k  +  1 ) )  <->  P  ||  A
) ) ) )
404, 8, 12, 16, 20, 39nnind 8438 . . 3  |-  ( N  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
4140impcom 123 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
)
42413impa 1138 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   1c1 7351    + caddc 7353    x. cmul 7355   NNcn 8422   NN0cn0 8673   ZZcz 8750   ^cexp 9954    || cdvds 11074   Primecprime 11367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-1o 6181  df-2o 6182  df-er 6292  df-en 6458  df-sup 6679  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-fl 9677  df-mod 9730  df-iseq 9853  df-seq3 9854  df-exp 9955  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-dvds 11075  df-gcd 11217  df-prm 11368
This theorem is referenced by:  prmdvdsexpb  11406  rpexp  11410
  Copyright terms: Public domain W3C validator