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Theorem pcdvdsb 12937
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 5882 . . . 4  |-  ( N  =  0  ->  ( P  pCnt  N )  =  ( P  pCnt  0
) )
21breq2d 4051 . . 3  |-  ( N  =  0  ->  ( A  <_  ( P  pCnt  N )  <->  A  <_  ( P 
pCnt  0 ) ) )
3 breq2 4043 . . 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 10131 . . . . . 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 12917 . . . . . . . 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 10131 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  ZZ )
13 eluz 10257 . . . . . 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 12777 . . . . . . . 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 10132 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  ZZ )
18 dvdsexp 12600 . . . . . . 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 12931 . . . . . 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 11132 . . . . . . . . . 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 10132 . . . . . . 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 11282 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  NN )
3029nnzd 10132 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  ZZ )
31 dvdstr 12578 . . . . . 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 9990 . . . . . . 7  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( P  pCnt  N )  e.  RR )
36 nn0re 9990 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
37 ltnle 8918 . . . . . . 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 10090 . . . . . 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 10020 . . . . . . . . 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 10131 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  ZZ )
45 eluz 10257 . . . . . . 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 12600 . . . . . . . 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 12933 . . . . . . . 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 11282 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  NN )
5453nnzd 10132 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ )
55 dvdstr 12578 . . . . . . . 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 8897 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR* )
65 pnfge 10485 . . . . 5  |-  ( A  e.  RR*  ->  A  <_  +oo )
6664, 65syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  +oo )
67 pc0 12923 . . . . 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 4065 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  ( P  pCnt  0
) )
70 dvds0 12560 . . . 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 2534 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcelnn  12938  pcidlem  12940  pcdvdstr  12944  pcgcd1  12945  pcfac  12963  pockthlem  12968  pockthg  12969  prmreclem2  12980  sylow1lem1  14925  sylow1lem3  14927  sylow1lem5  14929  ablfac1c  15322  ablfac1eu  15324  issqf  20390  vmasum  20471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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