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Theorem prmdvdsexp 11733
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 5748 . . . . . . 7  |-  ( m  =  1  ->  ( A ^ m )  =  ( A ^ 1 ) )
21breq2d 3909 . . . . . 6  |-  ( m  =  1  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ 1 ) ) )
32bibi1d 232 . . . . 5  |-  ( m  =  1  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ 1 )  <-> 
P  ||  A )
) )
43imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( m  =  k  ->  ( A ^ m )  =  ( A ^ k
) )
65breq2d 3909 . . . . . 6  |-  ( m  =  k  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ k ) ) )
76bibi1d 232 . . . . 5  |-  ( m  =  k  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
) )
87imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  ( A ^ m )  =  ( A ^ (
k  +  1 ) ) )
109breq2d 3909 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ ( k  +  1 ) ) ) )
1110bibi1d 232 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  A )
) )
1211imbi2d 229 . . . 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 5748 . . . . . . 7  |-  ( m  =  N  ->  ( A ^ m )  =  ( A ^ N
) )
1413breq2d 3909 . . . . . 6  |-  ( m  =  N  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ N ) ) )
1514bibi1d 232 . . . . 5  |-  ( m  =  N  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
1615imbi2d 229 . . . 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 9013 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1817adantl 273 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  CC )
1918exp1d 10370 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ 1 )  =  A )
2019breq2d 3909 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) )
21 nnnn0 8938 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
22 expp1 10251 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2318, 21, 22syl2an 285 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2423breq2d 3909 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  ( ( A ^ k )  x.  A ) ) )
25 simpll 501 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  P  e.  Prime )
26 simpr 109 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
27 zexpcl 10259 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
2826, 21, 27syl2an 285 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
k )  e.  ZZ )
29 simplr 502 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  A  e.  ZZ )
30 euclemma 11731 . . . . . . . . 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 1199 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( ( A ^
k )  x.  A
)  <->  ( P  ||  ( A ^ k )  \/  P  ||  A
) ) )
3224, 31bitrd 187 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
( P  ||  ( A ^ k )  \/  P  ||  A ) ) )
33 orbi1 764 . . . . . . . . 9  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  ( P  ||  A  \/  P  ||  A ) ) )
34 oridm 729 . . . . . . . . 9  |-  ( ( P  ||  A  \/  P  ||  A )  <->  P  ||  A
)
3533, 34syl6bb 195 . . . . . . . 8  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  P  ||  A
) )
3635bibi2d 231 . . . . . . 7  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
( k  +  1 ) )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) )  <->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3732, 36syl5ibcom 154 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3837expcom 115 . . . . 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 8696 . . 3  |-  ( N  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
4140impcom 124 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
)
42413impa 1159 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 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   CCcc 7582   1c1 7585    + caddc 7587    x. cmul 7589   NNcn 8680   NN0cn0 8931   ZZcz 9008   ^cexp 10243    || cdvds 11400   Primecprime 11695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-1o 6279  df-2o 6280  df-er 6395  df-en 6601  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8396  df-inn 8681  df-2 8739  df-3 8740  df-4 8741  df-n0 8932  df-z 9009  df-uz 9279  df-q 9364  df-rp 9394  df-fz 9742  df-fzo 9871  df-fl 9994  df-mod 10047  df-seqfrec 10170  df-exp 10244  df-cj 10565  df-re 10566  df-im 10567  df-rsqrt 10721  df-abs 10722  df-dvds 11401  df-gcd 11543  df-prm 11696
This theorem is referenced by:  prmdvdsexpb  11734  rpexp  11738
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