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Theorem prmdvdsexp 12809
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 5882 . . . . . . 7  |-  ( m  =  1  ->  ( A ^ m )  =  ( A ^ 1 ) )
21breq2d 4051 . . . . . 6  |-  ( m  =  1  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ 1 ) ) )
32bibi1d 310 . . . . 5  |-  ( m  =  1  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ 1 )  <-> 
P  ||  A )
) )
43imbi2d 307 . . . 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 5882 . . . . . . 7  |-  ( m  =  k  ->  ( A ^ m )  =  ( A ^ k
) )
65breq2d 4051 . . . . . 6  |-  ( m  =  k  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ k ) ) )
76bibi1d 310 . . . . 5  |-  ( m  =  k  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
) )
87imbi2d 307 . . . 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 5882 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  ( A ^ m )  =  ( A ^ (
k  +  1 ) ) )
109breq2d 4051 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ ( k  +  1 ) ) ) )
1110bibi1d 310 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  A )
) )
1211imbi2d 307 . . . 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 5882 . . . . . . 7  |-  ( m  =  N  ->  ( A ^ m )  =  ( A ^ N
) )
1413breq2d 4051 . . . . . 6  |-  ( m  =  N  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ N ) ) )
1514bibi1d 310 . . . . 5  |-  ( m  =  N  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
1615imbi2d 307 . . . 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 10045 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1817adantl 452 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  CC )
1918exp1d 11256 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ 1 )  =  A )
2019breq2d 4051 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) )
21 nnnn0 9988 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
22 expp1 11126 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2318, 21, 22syl2an 463 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2423breq2d 4051 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  ( ( A ^ k )  x.  A ) ) )
25 simpll 730 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  P  e.  Prime )
26 simpr 447 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
27 zexpcl 11134 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
2826, 21, 27syl2an 463 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
k )  e.  ZZ )
29 simplr 731 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  A  e.  ZZ )
30 euclemma 12803 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( ( A ^
k )  x.  A
)  <->  ( P  ||  ( A ^ k )  \/  P  ||  A
) ) )
3224, 31bitrd 244 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
( P  ||  ( A ^ k )  \/  P  ||  A ) ) )
33 orbi1 686 . . . . . . . . 9  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  ( P  ||  A  \/  P  ||  A ) ) )
34 oridm 500 . . . . . . . . 9  |-  ( ( P  ||  A  \/  P  ||  A )  <->  P  ||  A
)
3533, 34syl6bb 252 . . . . . . . 8  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  P  ||  A
) )
3635bibi2d 309 . . . . . . 7  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
( k  +  1 ) )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) )  <->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3732, 36syl5ibcom 211 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3837expcom 424 . . . . 5  |-  ( k  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) ) )
3938a2d 23 . . . 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 9780 . . 3  |-  ( N  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
4140impcom 419 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
)
42413impa 1146 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120    || cdivides 12547   Primecprime 12774
This theorem is referenced by:  prmdvdsexpb  12810  rpexp  12815  pythagtriplem4  12888  lgsqr  20601  2sqlem3  20621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775
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