MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prmdvdsexp Structured version   Unicode version

Theorem prmdvdsexp 14462
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 6285 . . . . . . 7  |-  ( m  =  1  ->  ( A ^ m )  =  ( A ^ 1 ) )
21breq2d 4406 . . . . . 6  |-  ( m  =  1  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ 1 ) ) )
32bibi1d 317 . . . . 5  |-  ( m  =  1  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ 1 )  <-> 
P  ||  A )
) )
43imbi2d 314 . . . 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 6285 . . . . . . 7  |-  ( m  =  k  ->  ( A ^ m )  =  ( A ^ k
) )
65breq2d 4406 . . . . . 6  |-  ( m  =  k  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ k ) ) )
76bibi1d 317 . . . . 5  |-  ( m  =  k  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ k )  <-> 
P  ||  A )
) )
87imbi2d 314 . . . 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 6285 . . . . . . 7  |-  ( m  =  ( k  +  1 )  ->  ( A ^ m )  =  ( A ^ (
k  +  1 ) ) )
109breq2d 4406 . . . . . 6  |-  ( m  =  ( k  +  1 )  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ ( k  +  1 ) ) ) )
1110bibi1d 317 . . . . 5  |-  ( m  =  ( k  +  1 )  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  A )
) )
1211imbi2d 314 . . . 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 6285 . . . . . . 7  |-  ( m  =  N  ->  ( A ^ m )  =  ( A ^ N
) )
1413breq2d 4406 . . . . . 6  |-  ( m  =  N  ->  ( P  ||  ( A ^
m )  <->  P  ||  ( A ^ N ) ) )
1514bibi1d 317 . . . . 5  |-  ( m  =  N  ->  (
( P  ||  ( A ^ m )  <->  P  ||  A
)  <->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
1615imbi2d 314 . . . 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 10909 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  CC )
1817adantl 464 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  CC )
1918exp1d 12347 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( A ^ 1 )  =  A )
2019breq2d 4406 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^
1 )  <->  P  ||  A
) )
21 nnnn0 10842 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
22 expp1 12215 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ (
k  +  1 ) )  =  ( ( A ^ k )  x.  A ) )
2318, 21, 22syl2an 475 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
( k  +  1 ) )  =  ( ( A ^ k
)  x.  A ) )
2423breq2d 4406 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
P  ||  ( ( A ^ k )  x.  A ) ) )
25 simpll 752 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  P  e.  Prime )
26 simpr 459 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  A  e.  ZZ )
27 zexpcl 12223 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  ZZ )
2826, 21, 27syl2an 475 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( A ^
k )  e.  ZZ )
29 simplr 754 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  A  e.  ZZ )
30 euclemma 14456 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( ( A ^
k )  x.  A
)  <->  ( P  ||  ( A ^ k )  \/  P  ||  A
) ) )
3224, 31bitrd 253 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( P  ||  ( A ^ ( k  +  1 ) )  <-> 
( P  ||  ( A ^ k )  \/  P  ||  A ) ) )
33 orbi1 704 . . . . . . . . 9  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  ( P  ||  A  \/  P  ||  A ) ) )
34 oridm 512 . . . . . . . . 9  |-  ( ( P  ||  A  \/  P  ||  A )  <->  P  ||  A
)
3533, 34syl6bb 261 . . . . . . . 8  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
k )  \/  P  ||  A )  <->  P  ||  A
) )
3635bibi2d 316 . . . . . . 7  |-  ( ( P  ||  ( A ^ k )  <->  P  ||  A
)  ->  ( ( P  ||  ( A ^
( k  +  1 ) )  <->  ( P  ||  ( A ^ k
)  \/  P  ||  A ) )  <->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3732, 36syl5ibcom 220 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  k  e.  NN )  ->  ( ( P 
||  ( A ^
k )  <->  P  ||  A
)  ->  ( P  ||  ( A ^ (
k  +  1 ) )  <->  P  ||  A ) ) )
3837expcom 433 . . . . 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 10593 . . 3  |-  ( N  e.  NN  ->  (
( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
) )
4140impcom 428 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ )  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <-> 
P  ||  A )
)
42413impa 1192 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  N  e.  NN )  ->  ( P  ||  ( A ^ N )  <->  P  ||  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394  (class class class)co 6277   CCcc 9519   1c1 9522    + caddc 9524    x. cmul 9526   NNcn 10575   NN0cn0 10835   ZZcz 10904   ^cexp 12208    || cdvds 14193   Primecprime 14424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-dvds 14194  df-gcd 14352  df-prm 14425
This theorem is referenced by:  prmdvdsexpb  14463  rpexp  14468  pythagtriplem4  14550  lgsqr  24000  2sqlem3  24020  etransclem41  37407
  Copyright terms: Public domain W3C validator