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Theorem pcdvdsb 12921
Description:  P ^ A divides  N if and only if  A is at most the count of  P. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
pcdvdsb  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )

Proof of Theorem pcdvdsb
StepHypRef Expression
1 oveq2 5866 . . . 4  |-  ( N  =  0  ->  ( P  pCnt  N )  =  ( P  pCnt  0
) )
21breq2d 4035 . . 3  |-  ( N  =  0  ->  ( A  <_  ( P  pCnt  N )  <->  A  <_  ( P 
pCnt  0 ) ) )
3 breq2 4027 . . 3  |-  ( N  =  0  ->  (
( P ^ A
)  ||  N  <->  ( P ^ A )  ||  0
) )
42, 3bibi12d 312 . 2  |-  ( N  =  0  ->  (
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
)  <->  ( A  <_ 
( P  pCnt  0
)  <->  ( P ^ A )  ||  0
) ) )
5 simpl3 960 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  NN0 )
65nn0zd 10115 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  ZZ )
7 simpl1 958 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  Prime )
8 simpl2 959 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  e.  ZZ )
9 simpr 447 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  =/=  0 )
10 pczcl 12901 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  NN0 )
117, 8, 9, 10syl12anc 1180 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  NN0 )
1211nn0zd 10115 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  ZZ )
13 eluz 10241 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( P  pCnt  N )  e.  ZZ )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
146, 12, 13syl2anc 642 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
15 prmnn 12761 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
167, 15syl 15 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  NN )
1716nnzd 10116 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  ZZ )
18 dvdsexp 12584 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  NN0  /\  ( P  pCnt  N )  e.  ( ZZ>= `  A )
)  ->  ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) ) )
19183expia 1153 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  ->  ( P ^ A )  ||  ( P ^ ( P 
pCnt  N ) ) ) )
2017, 5, 19syl2anc 642 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  ->  ( P ^ A )  ||  ( P ^ ( P 
pCnt  N ) ) ) )
2114, 20sylbird 226 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  ( P ^ ( P  pCnt  N ) ) ) )
22 pczdvds 12915 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
237, 8, 9, 22syl12anc 1180 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
24 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( P  e.  NN  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  NN )
2515, 24sylan 457 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
26253adant2 974 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  NN )
2726nnzd 10116 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  ZZ )
2827adantr 451 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ A
)  e.  ZZ )
2916, 11nnexpcld 11266 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  NN )
3029nnzd 10116 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  ZZ )
31 dvdstr 12562 . . . . . 6  |-  ( ( ( P ^ A
)  e.  ZZ  /\  ( P ^ ( P 
pCnt  N ) )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) )  /\  ( P ^ ( P  pCnt  N ) )  ||  N
)  ->  ( P ^ A )  ||  N
) )
3228, 30, 8, 31syl3anc 1182 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P ^ A )  ||  ( P ^ ( P 
pCnt  N ) )  /\  ( P ^ ( P 
pCnt  N ) )  ||  N )  ->  ( P ^ A )  ||  N ) )
3323, 32mpan2d 655 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) )  ->  ( P ^ A )  ||  N ) )
3421, 33syld 40 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  N )
)
35 nn0re 9974 . . . . . . 7  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( P  pCnt  N )  e.  RR )
36 nn0re 9974 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
37 ltnle 8902 . . . . . . 7  |-  ( ( ( P  pCnt  N
)  e.  RR  /\  A  e.  RR )  ->  ( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
3835, 36, 37syl2an 463 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
39 nn0ltp1le 10074 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  ( ( P  pCnt  N )  +  1 )  <_  A
) )
4038, 39bitr3d 246 . . . . 5  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
4111, 5, 40syl2anc 642 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
42 peano2nn0 10004 . . . . . . . . 9  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( ( P  pCnt  N )  +  1 )  e.  NN0 )
4311, 42syl 15 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  NN0 )
4443nn0zd 10115 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  ZZ )
45 eluz 10241 . . . . . . 7  |-  ( ( ( ( P  pCnt  N )  +  1 )  e.  ZZ  /\  A  e.  ZZ )  ->  ( A  e.  ( ZZ>= `  ( ( P  pCnt  N )  +  1 ) )  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
4644, 6, 45syl2anc 642 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  e.  (
ZZ>= `  ( ( P 
pCnt  N )  +  1 ) )  <->  ( ( P  pCnt  N )  +  1 )  <_  A
) )
47 dvdsexp 12584 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( P  pCnt  N )  +  1 )  e.  NN0  /\  A  e.  ( ZZ>= `  ( ( P  pCnt  N )  +  1 ) ) )  ->  ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A ) )
48473expia 1153 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( P  pCnt  N )  +  1 )  e.  NN0 )  -> 
( A  e.  (
ZZ>= `  ( ( P 
pCnt  N )  +  1 ) )  ->  ( P ^ ( ( P 
pCnt  N )  +  1 ) )  ||  ( P ^ A ) ) )
4917, 43, 48syl2anc 642 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  e.  (
ZZ>= `  ( ( P 
pCnt  N )  +  1 ) )  ->  ( P ^ ( ( P 
pCnt  N )  +  1 ) )  ||  ( P ^ A ) ) )
5046, 49sylbird 226 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P 
pCnt  N )  +  1 )  <_  A  ->  ( P ^ ( ( P  pCnt  N )  +  1 ) ) 
||  ( P ^ A ) ) )
51 pczndvds 12917 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
527, 8, 9, 51syl12anc 1180 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
5316, 43nnexpcld 11266 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  NN )
5453nnzd 10116 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ )
55 dvdstr 12562 . . . . . . . 8  |-  ( ( ( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ  /\  ( P ^ A )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N ) )
5654, 28, 8, 55syl3anc 1182 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P ^ ( ( P 
pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N ) )
5752, 56mtod 168 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
58 imnan 411 . . . . . 6  |-  ( ( ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )  <->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
5957, 58sylibr 203 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )
)
6050, 59syld 40 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P 
pCnt  N )  +  1 )  <_  A  ->  -.  ( P ^ A
)  ||  N )
)
6141, 60sylbid 206 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  ->  -.  ( P ^ A )  ||  N ) )
6234, 61impcon4bid 196 . 2  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )
63363ad2ant3 978 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR )
6463rexrd 8881 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR* )
65 pnfge 10469 . . . . 5  |-  ( A  e.  RR*  ->  A  <_  +oo )
6664, 65syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  +oo )
67 pc0 12907 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = 
+oo )
68673ad2ant1 976 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P  pCnt  0 )  = 
+oo )
6966, 68breqtrrd 4049 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  ( P  pCnt  0
) )
70 dvds0 12544 . . . 4  |-  ( ( P ^ A )  e.  ZZ  ->  ( P ^ A )  ||  0 )
7127, 70syl 15 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  ||  0 )
7269, 712thd 231 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  0 )  <->  ( P ^ A )  ||  0
) )
734, 62, 72pm2.61ne 2521 1  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104    || cdivides 12531   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcelnn  12922  pcidlem  12924  pcdvdstr  12928  pcgcd1  12929  pcfac  12947  pockthlem  12952  pockthg  12953  prmreclem2  12964  sylow1lem1  14909  sylow1lem3  14911  sylow1lem5  14913  ablfac1c  15306  ablfac1eu  15308  issqf  20374  vmasum  20455
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  ax-pre-sup 8815
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-rmo 2551  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-int 3863  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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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