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Theorem elqaalem1 19715
Description: Lemma for elqaa 19718. The function  N represents the denominators of the rational coefficients 
B. By multiplying them all together to make  R, we get a number big enough to clear all the denominators and make  R  x.  F an integer polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.)
Hypotheses
Ref Expression
elqaa.1  |-  ( ph  ->  A  e.  CC )
elqaa.2  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0 p } ) )
elqaa.3  |-  ( ph  ->  ( F `  A
)  =  0 )
elqaa.4  |-  B  =  (coeff `  F )
elqaa.5  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
elqaa.6  |-  R  =  (  seq  0 (  x.  ,  N ) `
 (deg `  F
) )
Assertion
Ref Expression
elqaalem1  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
Distinct variable groups:    k, n, A    B, k, n    ph, k    k, K, n    k, N, n    R, k
Allowed substitution hints:    ph( n)    R( n)    F( k, n)

Proof of Theorem elqaalem1
StepHypRef Expression
1 fveq2 5541 . . . . . . . . 9  |-  ( k  =  K  ->  ( B `  k )  =  ( B `  K ) )
21oveq1d 5889 . . . . . . . 8  |-  ( k  =  K  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 K )  x.  n ) )
32eleq1d 2362 . . . . . . 7  |-  ( k  =  K  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  n )  e.  ZZ ) )
43rabbidv 2793 . . . . . 6  |-  ( k  =  K  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
54supeq1d 7215 . . . . 5  |-  ( k  =  K  ->  sup ( { n  e.  NN  |  ( ( B `
 k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  =  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
6 elqaa.5 . . . . 5  |-  N  =  ( k  e.  NN0  |->  sup ( { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
7 ltso 8919 . . . . . . 7  |-  <  Or  RR
8 cnvso 5230 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
97, 8mpbi 199 . . . . . 6  |-  `'  <  Or  RR
109supex 7230 . . . . 5  |-  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
115, 6, 10fvmpt 5618 . . . 4  |-  ( K  e.  NN0  ->  ( N `
 K )  =  sup ( { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
1211adantl 452 . . 3  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  =  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
13 ssrab2 3271 . . . . 5  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  NN
14 nnuz 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
1513, 14sseqtri 3223 . . . 4  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
16 elqaa.2 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0 p } ) )
17 eldifi 3311 . . . . . . . . 9  |-  ( F  e.  ( (Poly `  QQ )  \  { 0 p } )  ->  F  e.  (Poly `  QQ ) )
1816, 17syl 15 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
19 0z 10051 . . . . . . . . 9  |-  0  e.  ZZ
20 zq 10338 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
2119, 20ax-mp 8 . . . . . . . 8  |-  0  e.  QQ
22 elqaa.4 . . . . . . . . 9  |-  B  =  (coeff `  F )
2322coef2 19629 . . . . . . . 8  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
2418, 21, 23sylancl 643 . . . . . . 7  |-  ( ph  ->  B : NN0 --> QQ )
25 ffvelrn 5679 . . . . . . 7  |-  ( ( B : NN0 --> QQ  /\  K  e.  NN0 )  -> 
( B `  K
)  e.  QQ )
2624, 25sylan 457 . . . . . 6  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( B `  K )  e.  QQ )
27 qmulz 10335 . . . . . 6  |-  ( ( B `  K )  e.  QQ  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
2826, 27syl 15 . . . . 5  |-  ( (
ph  /\  K  e.  NN0 )  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
29 rabn0 3487 . . . . 5  |-  ( { n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }  =/=  (/)  <->  E. n  e.  NN  ( ( B `  K )  x.  n
)  e.  ZZ )
3028, 29sylibr 203 . . . 4  |-  ( (
ph  /\  K  e.  NN0 )  ->  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  =/=  (/) )
31 infmssuzcl 10317 . . . 4  |-  ( ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ }  C_  ( ZZ>= `  1
)  /\  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  =/=  (/) )  ->  sup ( { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e. 
{ n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } )
3215, 30, 31sylancr 644 . . 3  |-  ( (
ph  /\  K  e.  NN0 )  ->  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
3312, 32eqeltrd 2370 . 2  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  e.  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
34 oveq2 5882 . . . 4  |-  ( n  =  ( N `  K )  ->  (
( B `  K
)  x.  n )  =  ( ( B `
 K )  x.  ( N `  K
) ) )
3534eleq1d 2362 . . 3  |-  ( n  =  ( N `  K )  ->  (
( ( B `  K )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  ( N `  K
) )  e.  ZZ ) )
3635elrab 2936 . 2  |-  ( ( N `  K )  e.  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  <->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
3733, 36sylib 188 1  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093    Or wor 4329   `'ccnv 4704   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   QQcq 10332    seq cseq 11062   0 pc0p 19040  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  elqaalem2  19716
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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  ax-addf 8832
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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  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-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588
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