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

Theorem pcdvdsb 14250
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 6291 . . . 4  |-  ( N  =  0  ->  ( P  pCnt  N )  =  ( P  pCnt  0
) )
21breq2d 4459 . . 3  |-  ( N  =  0  ->  ( A  <_  ( P  pCnt  N )  <->  A  <_  ( P 
pCnt  0 ) ) )
3 breq2 4451 . . 3  |-  ( N  =  0  ->  (
( P ^ A
)  ||  N  <->  ( P ^ A )  ||  0
) )
42, 3bibi12d 321 . 2  |-  ( N  =  0  ->  (
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
)  <->  ( A  <_ 
( P  pCnt  0
)  <->  ( P ^ A )  ||  0
) ) )
5 simpl3 1001 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  NN0 )
65nn0zd 10963 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  ZZ )
7 simpl1 999 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  Prime )
8 simpl2 1000 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  e.  ZZ )
9 simpr 461 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  N  =/=  0 )
10 pczcl 14230 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  NN0 )
117, 8, 9, 10syl12anc 1226 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  NN0 )
1211nn0zd 10963 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  ZZ )
13 eluz 11094 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( P  pCnt  N )  e.  ZZ )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
146, 12, 13syl2anc 661 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  <->  A  <_  ( P  pCnt  N )
) )
15 prmnn 14078 . . . . . . . 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 10964 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  ZZ )
18 dvdsexp 13900 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  NN0  /\  ( P  pCnt  N )  e.  ( ZZ>= `  A )
)  ->  ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) ) )
19183expia 1198 . . . . . 6  |-  ( ( P  e.  ZZ  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  e.  ( ZZ>= `  A )  ->  ( P ^ A )  ||  ( P ^ ( P 
pCnt  N ) ) ) )
2017, 5, 19syl2anc 661 . . . . 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 235 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  ( P ^ ( P  pCnt  N ) ) ) )
22 pczdvds 14244 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
237, 8, 9, 22syl12anc 1226 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
24 nnexpcl 12146 . . . . . . . . . 10  |-  ( ( P  e.  NN  /\  A  e.  NN0 )  -> 
( P ^ A
)  e.  NN )
2515, 24sylan 471 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
26253adant2 1015 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  NN )
2726nnzd 10964 . . . . . . 7  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  ZZ )
2827adantr 465 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ A
)  e.  ZZ )
2916, 11nnexpcld 12298 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  NN )
3029nnzd 10964 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  ZZ )
31 dvdstr 13877 . . . . . 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 1228 . . . . 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 674 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) )  ->  ( P ^ A )  ||  N ) )
3421, 33syld 44 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  -> 
( P ^ A
)  ||  N )
)
35 nn0re 10803 . . . . . . 7  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( P  pCnt  N )  e.  RR )
36 nn0re 10803 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
37 ltnle 9663 . . . . . . 7  |-  ( ( ( P  pCnt  N
)  e.  RR  /\  A  e.  RR )  ->  ( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
3835, 36, 37syl2an 477 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  -.  A  <_  ( P  pCnt  N
) ) )
39 nn0ltp1le 10919 . . . . . 6  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( ( P  pCnt  N )  <  A  <->  ( ( P  pCnt  N )  +  1 )  <_  A
) )
4038, 39bitr3d 255 . . . . 5  |-  ( ( ( P  pCnt  N
)  e.  NN0  /\  A  e.  NN0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
4111, 5, 40syl2anc 661 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  <->  ( ( P 
pCnt  N )  +  1 )  <_  A )
)
42 peano2nn0 10835 . . . . . . . . 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 10963 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  ZZ )
45 eluz 11094 . . . . . . 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 661 . . . . . 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 13900 . . . . . . . 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 1198 . . . . . . 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 661 . . . . . 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 235 . . . . 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 14246 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
527, 8, 9, 51syl12anc 1226 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
5316, 43nnexpcld 12298 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  NN )
5453nnzd 10964 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ )
55 dvdstr 13877 . . . . . . . 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 1228 . . . . . . 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 177 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
58 imnan 422 . . . . . 6  |-  ( ( ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )  <->  -.  ( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  /\  ( P ^ A ) 
||  N ) )
5957, 58sylibr 212 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P ^
( ( P  pCnt  N )  +  1 ) )  ||  ( P ^ A )  ->  -.  ( P ^ A
)  ||  N )
)
6050, 59syld 44 . . . 4  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( ( P 
pCnt  N )  +  1 )  <_  A  ->  -.  ( P ^ A
)  ||  N )
)
6141, 60sylbid 215 . . 3  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( -.  A  <_ 
( P  pCnt  N
)  ->  -.  ( P ^ A )  ||  N ) )
6234, 61impcon4bid 205 . 2  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )
63363ad2ant3 1019 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR )
6463rexrd 9642 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR* )
65 pnfge 11338 . . . . 5  |-  ( A  e.  RR*  ->  A  <_ +oo )
6664, 65syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_ +oo )
67 pc0 14236 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = +oo )
68673ad2ant1 1017 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P  pCnt  0 )  = +oo )
6966, 68breqtrrd 4473 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  ( P  pCnt  0
) )
70 dvds0 13859 . . . 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 240 . 2  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  0 )  <->  ( P ^ A )  ||  0
) )
734, 62, 72pm2.61ne 2782 1  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  N )  <->  ( P ^ A )  ||  N
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494   +oocpnf 9624   RR*cxr 9626    < clt 9627    <_ cle 9628   NNcn 10535   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ^cexp 12133    || cdivides 13846   Primecprime 14075    pCnt cpc 14218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-dvds 13847  df-gcd 14003  df-prm 14076  df-pc 14219
This theorem is referenced by:  pcelnn  14251  pcidlem  14253  pcdvdstr  14257  pcgcd1  14258  pcfac  14276  pockthlem  14281  pockthg  14282  prmreclem2  14293  sylow1lem1  16421  sylow1lem3  16423  sylow1lem5  16425  ablfac1c  16921  ablfac1eu  16923  issqf  23154  vmasum  23235
  Copyright terms: Public domain W3C validator