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Theorem pcprendvds2 13927
Description: Non-divisibility property of the prime power pre-function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Hypotheses
Ref Expression
pclem.1  |-  A  =  { n  e.  NN0  |  ( P ^ n
)  ||  N }
pclem.2  |-  S  =  sup ( A ,  RR ,  <  )
Assertion
Ref Expression
pcprendvds2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ S ) ) )
Distinct variable groups:    n, N    P, n
Allowed substitution hints:    A( n)    S( n)

Proof of Theorem pcprendvds2
StepHypRef Expression
1 pclem.1 . . 3  |-  A  =  { n  e.  NN0  |  ( P ^ n
)  ||  N }
2 pclem.2 . . 3  |-  S  =  sup ( A ,  RR ,  <  )
31, 2pcprendvds 13926 . 2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  ( P ^ ( S  +  1 ) )  ||  N )
4 eluz2b2 10946 . . . . . . 7  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  NN  /\  1  < 
P ) )
54simplbi 460 . . . . . 6  |-  ( P  e.  ( ZZ>= `  2
)  ->  P  e.  NN )
65adantr 465 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  NN )
76nnzd 10765 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  ZZ )
81, 2pcprecl 13925 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
98simprd 463 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  ||  N )
108simpld 459 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
116, 10nnexpcld 12048 . . . . . . 7  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  NN )
1211nnzd 10765 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  ZZ )
1311nnne0d 10385 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  =/=  0 )
14 simprl 755 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
15 dvdsval2 13557 . . . . . 6  |-  ( ( ( P ^ S
)  e.  ZZ  /\  ( P ^ S )  =/=  0  /\  N  e.  ZZ )  ->  (
( P ^ S
)  ||  N  <->  ( N  /  ( P ^ S ) )  e.  ZZ ) )
1612, 13, 14, 15syl3anc 1218 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  ||  N  <->  ( N  /  ( P ^ S ) )  e.  ZZ ) )
179, 16mpbid 210 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  ( P ^ S ) )  e.  ZZ )
18 dvdscmul 13578 . . . 4  |-  ( ( P  e.  ZZ  /\  ( N  /  ( P ^ S ) )  e.  ZZ  /\  ( P ^ S )  e.  ZZ )  ->  ( P  ||  ( N  / 
( P ^ S
) )  ->  (
( P ^ S
)  x.  P ) 
||  ( ( P ^ S )  x.  ( N  /  ( P ^ S ) ) ) ) )
197, 17, 12, 18syl3anc 1218 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  ||  ( N  /  ( P ^ S ) )  -> 
( ( P ^ S )  x.  P
)  ||  ( ( P ^ S )  x.  ( N  /  ( P ^ S ) ) ) ) )
206nncnd 10357 . . . . . 6  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  P  e.  CC )
2120, 10expp1d 12028 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ ( S  +  1 ) )  =  ( ( P ^ S )  x.  P ) )
2221eqcomd 2448 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  x.  P
)  =  ( P ^ ( S  + 
1 ) ) )
23 zcn 10670 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
2423ad2antrl 727 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
2511nncnd 10357 . . . . 5  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P ^ S
)  e.  CC )
2624, 25, 13divcan2d 10128 . . . 4  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( P ^ S )  x.  ( N  /  ( P ^ S ) ) )  =  N )
2722, 26breq12d 4324 . . 3  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( P ^ S )  x.  P )  ||  (
( P ^ S
)  x.  ( N  /  ( P ^ S ) ) )  <-> 
( P ^ ( S  +  1 ) )  ||  N ) )
2819, 27sylibd 214 . 2  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  ||  ( N  /  ( P ^ S ) )  -> 
( P ^ ( S  +  1 ) )  ||  N ) )
293, 28mtod 177 1  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  -.  P  ||  ( N  /  ( P ^ S ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2620   {crab 2738   class class class wbr 4311   ` cfv 5437  (class class class)co 6110   supcsup 7709   CCcc 9299   RRcr 9300   0cc0 9301   1c1 9302    + caddc 9304    x. cmul 9306    < clt 9437    / cdiv 10012   NNcn 10341   2c2 10390   NN0cn0 10598   ZZcz 10665   ZZ>=cuz 10880   ^cexp 11884    || cdivides 13554
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 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
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 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-2nd 6597  df-recs 6851  df-rdg 6885  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-sup 7710  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fl 11661  df-seq 11826  df-exp 11885  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-dvds 13555
This theorem is referenced by:  pcpremul  13929  pczndvds2  13952
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