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Theorem pcdvdsb 13947
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 6111 . . . 4  |-  ( N  =  0  ->  ( P  pCnt  N )  =  ( P  pCnt  0
) )
21breq2d 4316 . . 3  |-  ( N  =  0  ->  ( A  <_  ( P  pCnt  N )  <->  A  <_  ( P 
pCnt  0 ) ) )
3 breq2 4308 . . 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 993 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  NN0 )
65nn0zd 10757 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  A  e.  ZZ )
7 simpl1 991 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  Prime )
8 simpl2 992 . . . . . . . 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 13927 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  NN0 )
117, 8, 9, 10syl12anc 1216 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  NN0 )
1211nn0zd 10757 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P  pCnt  N
)  e.  ZZ )
13 eluz 10886 . . . . . 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 13778 . . . . . . . 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 10758 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  P  e.  ZZ )
18 dvdsexp 13601 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  A  e.  NN0  /\  ( P  pCnt  N )  e.  ( ZZ>= `  A )
)  ->  ( P ^ A )  ||  ( P ^ ( P  pCnt  N ) ) )
19183expia 1189 . . . . . 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 13941 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
237, 8, 9, 22syl12anc 1216 . . . . 5  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) ) 
||  N )
24 nnexpcl 11890 . . . . . . . . . 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 1007 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P ^ A )  e.  NN )
2726nnzd 10758 . . . . . . 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 12041 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  NN )
3029nnzd 10758 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ ( P  pCnt  N ) )  e.  ZZ )
31 dvdstr 13578 . . . . . 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 1218 . . . . 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 10600 . . . . . . 7  |-  ( ( P  pCnt  N )  e.  NN0  ->  ( P  pCnt  N )  e.  RR )
36 nn0re 10600 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  RR )
37 ltnle 9466 . . . . . . 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 10714 . . . . . 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 10632 . . . . . . . . 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 10757 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( ( P  pCnt  N )  +  1 )  e.  ZZ )
45 eluz 10886 . . . . . . 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 13601 . . . . . . . 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 1189 . . . . . . 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 13943 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
527, 8, 9, 51syl12anc 1216 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  ->  -.  ( P ^ (
( P  pCnt  N
)  +  1 ) )  ||  N )
5316, 43nnexpcld 12041 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  NN )
5453nnzd 10758 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e.  NN0 )  /\  N  =/=  0 )  -> 
( P ^ (
( P  pCnt  N
)  +  1 ) )  e.  ZZ )
55 dvdstr 13578 . . . . . . . 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 1218 . . . . . . 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 1011 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR )
6463rexrd 9445 . . . . 5  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  e.  RR* )
65 pnfge 11122 . . . . 5  |-  ( A  e.  RR*  ->  A  <_ +oo )
6664, 65syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_ +oo )
67 pc0 13933 . . . . 5  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = +oo )
68673ad2ant1 1009 . . . 4  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  ( P  pCnt  0 )  = +oo )
6966, 68breqtrrd 4330 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  ZZ  /\  A  e. 
NN0 )  ->  A  <_  ( P  pCnt  0
) )
70 dvds0 13560 . . . 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 2698 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297   +oocpnf 9427   RR*cxr 9429    < clt 9430    <_ cle 9431   NNcn 10334   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   ^cexp 11877    || cdivides 13547   Primecprime 13775    pCnt cpc 13915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-gcd 13703  df-prm 13776  df-pc 13916
This theorem is referenced by:  pcelnn  13948  pcidlem  13950  pcdvdstr  13954  pcgcd1  13955  pcfac  13973  pockthlem  13978  pockthg  13979  prmreclem2  13990  sylow1lem1  16109  sylow1lem3  16111  sylow1lem5  16113  ablfac1c  16584  ablfac1eu  16586  issqf  22486  vmasum  22567
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