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Theorem elqaalem1 21784
Description: Lemma for elqaa 21787. 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 )  \  {
0p } ) )
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 5690 . . . . . . . . 9  |-  ( k  =  K  ->  ( B `  k )  =  ( B `  K ) )
21oveq1d 6105 . . . . . . . 8  |-  ( k  =  K  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 K )  x.  n ) )
32eleq1d 2508 . . . . . . 7  |-  ( k  =  K  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  n )  e.  ZZ ) )
43rabbidv 2963 . . . . . 6  |-  ( k  =  K  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
54supeq1d 7695 . . . . 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 gtso 9455 . . . . . 6  |-  `'  <  Or  RR
87supex 7712 . . . . 5  |-  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
95, 6, 8fvmpt 5773 . . . 4  |-  ( K  e.  NN0  ->  ( N `
 K )  =  sup ( { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
109adantl 466 . . 3  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  =  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
11 ssrab2 3436 . . . . 5  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  NN
12 nnuz 10895 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
1311, 12sseqtri 3387 . . . 4  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
14 elqaa.2 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0p } ) )
1514eldifad 3339 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
16 0z 10656 . . . . . . . . 9  |-  0  e.  ZZ
17 zq 10958 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
1816, 17ax-mp 5 . . . . . . . 8  |-  0  e.  QQ
19 elqaa.4 . . . . . . . . 9  |-  B  =  (coeff `  F )
2019coef2 21698 . . . . . . . 8  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
2115, 18, 20sylancl 662 . . . . . . 7  |-  ( ph  ->  B : NN0 --> QQ )
2221ffvelrnda 5842 . . . . . 6  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( B `  K )  e.  QQ )
23 qmulz 10955 . . . . . 6  |-  ( ( B `  K )  e.  QQ  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
2422, 23syl 16 . . . . 5  |-  ( (
ph  /\  K  e.  NN0 )  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
25 rabn0 3656 . . . . 5  |-  ( { n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }  =/=  (/)  <->  E. n  e.  NN  ( ( B `  K )  x.  n
)  e.  ZZ )
2624, 25sylibr 212 . . . 4  |-  ( (
ph  /\  K  e.  NN0 )  ->  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  =/=  (/) )
27 infmssuzcl 10937 . . . 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 } )
2813, 26, 27sylancr 663 . . 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 }
)
2910, 28eqeltrd 2516 . 2  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  e.  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
30 oveq2 6098 . . . 4  |-  ( n  =  ( N `  K )  ->  (
( B `  K
)  x.  n )  =  ( ( B `
 K )  x.  ( N `  K
) ) )
3130eleq1d 2508 . . 3  |-  ( n  =  ( N `  K )  ->  (
( ( B `  K )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  ( N `  K
) )  e.  ZZ ) )
3231elrab 3116 . 2  |-  ( ( N `  K )  e.  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  <->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
3329, 32sylib 196 1  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715   {crab 2718    \ cdif 3324    C_ wss 3327   (/)c0 3636   {csn 3876    e. cmpt 4349   `'ccnv 4838   -->wf 5413   ` cfv 5417  (class class class)co 6090   supcsup 7689   CCcc 9279   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    < clt 9417   NNcn 10321   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   QQcq 10952    seqcseq 11805   0pc0p 21146  Polycply 21651  coeffccoe 21653  degcdgr 21654
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-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147  df-ply 21655  df-coe 21657
This theorem is referenced by:  elqaalem2  21785
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