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Theorem pcdvdsb 13242
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 6089 . . . 4  |-  ( N  =  0  ->  ( P  pCnt  N )  =  ( P  pCnt  0
) )
21breq2d 4224 . . 3  |-  ( N  =  0  ->  ( A  <_  ( P  pCnt  N )  <->  A  <_  ( P 
pCnt  0 ) ) )
3 breq2 4216 . . 3  |-  ( N  =  0  ->  (
( P ^ A
)  ||  N  <->  ( P ^ A )  ||  0
) )
42, 3bibi12d 313 . 2  |-  ( N  =  0  ->  (
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
)  <->  ( A  <_ 
( P  pCnt  0
)  <->  ( P ^ A )  ||  0
) ) )
5 simpl3 962 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  NN0 )
65nn0zd 10373 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  ZZ )
7 simpl1 960 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  Prime )
8 simpl2 961 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  e.  ZZ )
9 simpr 448 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  =/=  0 )
10 pczcl 13222 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  NN0 )
117, 8, 9, 10syl12anc 1182 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  NN0 )
1211nn0zd 10373 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  ZZ )
13 eluz 10499 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( P  pCnt  N )  e.  ZZ )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
146, 12, 13syl2anc 643 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
15 prmnn 13082 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
167, 15syl 16 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  NN )
1716nnzd 10374 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  ZZ )
18 dvdsexp 12905 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  NN0  /\  ( P  pCnt  N )  e.  ( ZZ>= `  A )
)  ->  ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) ) )
19183expia 1155 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  ->  ( P ^ A )  ||  ( P ^ ( P 
pCnt  N ) ) ) )
2017, 5, 19syl2anc 643 . . . . 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 227 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  ( P ^ ( P  pCnt  N ) ) ) )
22 pczdvds 13236 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
237, 8, 9, 22syl12anc 1182 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
24 nnexpcl 11394 . . . . . . . . . 10  |-  ( ( P  e.  NN  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  NN )
2515, 24sylan 458 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
26253adant2 976 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  NN )
2726nnzd 10374 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  ZZ )
2827adantr 452 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ A
)  e.  ZZ )
2916, 11nnexpcld 11544 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  NN )
3029nnzd 10374 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  ZZ )
31 dvdstr 12883 . . . . . 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 1184 . . . . 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 656 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) )  ->  ( P ^ A )  ||  N ) )
3421, 33syld 42 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  N )
)
35 nn0re 10230 . . . . . . 7  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( P  pCnt  N )  e.  RR )
36 nn0re 10230 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
37 ltnle 9155 . . . . . . 7  |-  ( ( ( P  pCnt  N
)  e.  RR  /\  A  e.  RR )  ->  ( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
3835, 36, 37syl2an 464 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
39 nn0ltp1le 10332 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  ( ( P  pCnt  N )  +  1 )  <_  A
) )
4038, 39bitr3d 247 . . . . 5  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
4111, 5, 40syl2anc 643 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
42 peano2nn0 10260 . . . . . . . . 9  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( ( P  pCnt  N )  +  1 )  e.  NN0 )
4311, 42syl 16 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  NN0 )
4443nn0zd 10373 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  ZZ )
45 eluz 10499 . . . . . . 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 643 . . . . . 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 12905 . . . . . . . 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 1155 . . . . . . 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 643 . . . . . 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 227 . . . . 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 13238 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
527, 8, 9, 51syl12anc 1182 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
5316, 43nnexpcld 11544 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  NN )
5453nnzd 10374 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ )
55 dvdstr 12883 . . . . . . . 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 1184 . . . . . . 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 170 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
58 imnan 412 . . . . . 6  |-  ( ( ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )  <->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
5957, 58sylibr 204 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )
)
6050, 59syld 42 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P 
pCnt  N )  +  1 )  <_  A  ->  -.  ( P ^ A
)  ||  N )
)
6141, 60sylbid 207 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  ->  -.  ( P ^ A )  ||  N ) )
6234, 61impcon4bid 197 . 2  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )
63363ad2ant3 980 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR )
6463rexrd 9134 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR* )
65 pnfge 10727 . . . . 5  |-  ( A  e.  RR*  ->  A  <_  +oo )
6664, 65syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  +oo )
67 pc0 13228 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = 
+oo )
68673ad2ant1 978 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P  pCnt  0 )  = 
+oo )
6966, 68breqtrrd 4238 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  ( P  pCnt  0
) )
70 dvds0 12865 . . . 4  |-  ( ( P ^ A )  e.  ZZ  ->  ( P ^ A )  ||  0 )
7127, 70syl 16 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  ||  0 )
7269, 712thd 232 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  0 )  <->  ( P ^ A )  ||  0
) )
734, 62, 72pm2.61ne 2679 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ^cexp 11382    || cdivides 12852   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  pcelnn  13243  pcidlem  13245  pcdvdstr  13249  pcgcd1  13250  pcfac  13268  pockthlem  13273  pockthg  13274  prmreclem2  13285  sylow1lem1  15232  sylow1lem3  15234  sylow1lem5  15236  ablfac1c  15629  ablfac1eu  15631  issqf  20919  vmasum  21000
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-1st 6349  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-q 10575  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  df-pc 13211
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