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Theorem dvdspw 24103
Description: Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dvdspw  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N
) ) )

Proof of Theorem dvdspw
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 divides 12533 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  ->  ( K  ||  M  <->  E. m  e.  ZZ  (
m  x.  K )  =  M ) )
213adant3 975 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  <->  E. m  e.  ZZ  ( m  x.  K )  =  M ) )
3 simpl1 958 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  K  e.  ZZ )
4 simpl3 960 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  N  e.  NN )
5 iddvdsexp 12552 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  N  e.  NN )  ->  K  ||  ( K ^ N ) )
63, 4, 5syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  K  ||  ( K ^ N ) )
7 simpr 447 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
8 nnnn0 9972 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  NN0 )
983ad2ant3 978 . . . . . . . . 9  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  N  e.  NN0 )
109adantr 451 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  N  e.  NN0 )
11 zexpcl 11118 . . . . . . . 8  |-  ( ( m  e.  ZZ  /\  N  e.  NN0 )  -> 
( m ^ N
)  e.  ZZ )
127, 10, 11syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( m ^ N )  e.  ZZ )
13 zexpcl 11118 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  N  e.  NN0 )  -> 
( K ^ N
)  e.  ZZ )
143, 10, 13syl2anc 642 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( K ^ N )  e.  ZZ )
15 dvdsmul2 12551 . . . . . . 7  |-  ( ( ( m ^ N
)  e.  ZZ  /\  ( K ^ N )  e.  ZZ )  -> 
( K ^ N
)  ||  ( (
m ^ N )  x.  ( K ^ N ) ) )
1612, 14, 15syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( K ^ N )  ||  (
( m ^ N
)  x.  ( K ^ N ) ) )
1712, 14zmulcld 10123 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( ( m ^ N )  x.  ( K ^ N
) )  e.  ZZ )
18 dvdstr 12562 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  ( K ^ N )  e.  ZZ  /\  (
( m ^ N
)  x.  ( K ^ N ) )  e.  ZZ )  -> 
( ( K  ||  ( K ^ N )  /\  ( K ^ N )  ||  (
( m ^ N
)  x.  ( K ^ N ) ) )  ->  K  ||  (
( m ^ N
)  x.  ( K ^ N ) ) ) )
193, 14, 17, 18syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( ( K 
||  ( K ^ N )  /\  ( K ^ N )  ||  ( ( m ^ N )  x.  ( K ^ N ) ) )  ->  K  ||  (
( m ^ N
)  x.  ( K ^ N ) ) ) )
206, 16, 19mp2and 660 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  K  ||  (
( m ^ N
)  x.  ( K ^ N ) ) )
21 zcn 10029 . . . . . . 7  |-  ( m  e.  ZZ  ->  m  e.  CC )
2221adantl 452 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  m  e.  CC )
23 zcn 10029 . . . . . . . 8  |-  ( K  e.  ZZ  ->  K  e.  CC )
24233ad2ant1 976 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  K  e.  CC )
2524adantr 451 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  K  e.  CC )
2622, 25, 10mulexpd 11260 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( ( m  x.  K ) ^ N )  =  ( ( m ^ N
)  x.  ( K ^ N ) ) )
2720, 26breqtrrd 4049 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  K  ||  (
( m  x.  K
) ^ N ) )
28 oveq1 5865 . . . . 5  |-  ( ( m  x.  K )  =  M  ->  (
( m  x.  K
) ^ N )  =  ( M ^ N ) )
2928breq2d 4035 . . . 4  |-  ( ( m  x.  K )  =  M  ->  ( K  ||  ( ( m  x.  K ) ^ N )  <->  K  ||  ( M ^ N ) ) )
3027, 29syl5ibcom 211 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  /\  m  e.  ZZ )  ->  ( ( m  x.  K )  =  M  ->  K  ||  ( M ^ N ) ) )
3130rexlimdva 2667 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( E. m  e.  ZZ  ( m  x.  K
)  =  M  ->  K  ||  ( M ^ N ) ) )
322, 31sylbid 206 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  NN )  ->  ( K  ||  M  ->  K  ||  ( M ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023  (class class class)co 5858   CCcc 8735    x. cmul 8742   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104    || cdivides 12531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105  df-dvds 12532
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