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Theorem prmdvdsexp 13114
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 6089 . . . . . . 7  |-  ( m  =  1  ->  ( A ^ m )  =  ( A ^ 1 ) )
21breq2d 4224 . . . . . 6  |-  ( m  =  1  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ 1 ) ) )
32bibi1d 311 . . . . 5  |-  ( m  =  1  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ 1 )  <-> 
P  ||  A )
) )
43imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( m  =  k  ->  ( A ^ m )  =  ( A ^ k
) )
65breq2d 4224 . . . . . 6  |-  ( m  =  k  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ k ) ) )
76bibi1d 311 . . . . 5  |-  ( m  =  k  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
) )
87imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  ( A ^ m )  =  ( A ^ (
k  +  1 ) ) )
109breq2d 4224 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ ( k  +  1 ) ) ) )
1110bibi1d 311 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  A )
) )
1211imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( m  =  N  ->  ( A ^ m )  =  ( A ^ N
) )
1413breq2d 4224 . . . . . 6  |-  ( m  =  N  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ N ) ) )
1514bibi1d 311 . . . . 5  |-  ( m  =  N  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
1615imbi2d 308 . . . 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 10287 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1817adantl 453 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  CC )
1918exp1d 11518 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ 1 )  =  A )
2019breq2d 4224 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) )
21 nnnn0 10228 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
22 expp1 11388 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2318, 21, 22syl2an 464 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2423breq2d 4224 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  ( ( A ^ k )  x.  A ) ) )
25 simpll 731 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  P  e.  Prime )
26 simpr 448 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
27 zexpcl 11396 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
2826, 21, 27syl2an 464 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
k )  e.  ZZ )
29 simplr 732 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  A  e.  ZZ )
30 euclemma 13108 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( ( A ^
k )  x.  A
)  <->  ( P  ||  ( A ^ k )  \/  P  ||  A
) ) )
3224, 31bitrd 245 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
( P  ||  ( A ^ k )  \/  P  ||  A ) ) )
33 orbi1 687 . . . . . . . . 9  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  ( P  ||  A  \/  P  ||  A ) ) )
34 oridm 501 . . . . . . . . 9  |-  ( ( P  ||  A  \/  P  ||  A )  <->  P  ||  A
)
3533, 34syl6bb 253 . . . . . . . 8  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  P  ||  A
) )
3635bibi2d 310 . . . . . . 7  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
( k  +  1 ) )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) )  <->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3732, 36syl5ibcom 212 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3837expcom 425 . . . . 5  |-  ( k  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) ) )
3938a2d 24 . . . 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 10018 . . 3  |-  ( N  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
4140impcom 420 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
)
42413impa 1148 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    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995   NNcn 10000   NN0cn0 10221   ZZcz 10282   ^cexp 11382    || cdivides 12852   Primecprime 13079
This theorem is referenced by:  prmdvdsexpb  13115  rpexp  13120  pythagtriplem4  13193  lgsqr  21130  2sqlem3  21150
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080
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