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Theorem elqaalem1 20236
Description: Lemma for elqaa 20239. 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 5728 . . . . . . . . 9  |-  ( k  =  K  ->  ( B `  k )  =  ( B `  K ) )
21oveq1d 6096 . . . . . . . 8  |-  ( k  =  K  ->  (
( B `  k
)  x.  n )  =  ( ( B `
 K )  x.  n ) )
32eleq1d 2502 . . . . . . 7  |-  ( k  =  K  ->  (
( ( B `  k )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  n )  e.  ZZ ) )
43rabbidv 2948 . . . . . 6  |-  ( k  =  K  ->  { n  e.  NN  |  ( ( B `  k )  x.  n )  e.  ZZ }  =  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
54supeq1d 7451 . . . . 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 9156 . . . . . . 7  |-  <  Or  RR
8 cnvso 5411 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
97, 8mpbi 200 . . . . . 6  |-  `'  <  Or  RR
109supex 7468 . . . . 5  |-  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  )  e.  _V
115, 6, 10fvmpt 5806 . . . 4  |-  ( K  e.  NN0  ->  ( N `
 K )  =  sup ( { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
1211adantl 453 . . 3  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  =  sup ( { n  e.  NN  |  ( ( B `
 K )  x.  n )  e.  ZZ } ,  RR ,  `'  <  ) )
13 ssrab2 3428 . . . . 5  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  NN
14 nnuz 10521 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
1513, 14sseqtri 3380 . . . 4  |-  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  C_  ( ZZ>=
`  1 )
16 elqaa.2 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( (Poly `  QQ )  \  {
0 p } ) )
1716eldifad 3332 . . . . . . . 8  |-  ( ph  ->  F  e.  (Poly `  QQ ) )
18 0z 10293 . . . . . . . . 9  |-  0  e.  ZZ
19 zq 10580 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
2018, 19ax-mp 8 . . . . . . . 8  |-  0  e.  QQ
21 elqaa.4 . . . . . . . . 9  |-  B  =  (coeff `  F )
2221coef2 20150 . . . . . . . 8  |-  ( ( F  e.  (Poly `  QQ )  /\  0  e.  QQ )  ->  B : NN0 --> QQ )
2317, 20, 22sylancl 644 . . . . . . 7  |-  ( ph  ->  B : NN0 --> QQ )
2423ffvelrnda 5870 . . . . . 6  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( B `  K )  e.  QQ )
25 qmulz 10577 . . . . . 6  |-  ( ( B `  K )  e.  QQ  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
2624, 25syl 16 . . . . 5  |-  ( (
ph  /\  K  e.  NN0 )  ->  E. n  e.  NN  ( ( B `
 K )  x.  n )  e.  ZZ )
27 rabn0 3647 . . . . 5  |-  ( { n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }  =/=  (/)  <->  E. n  e.  NN  ( ( B `  K )  x.  n
)  e.  ZZ )
2826, 27sylibr 204 . . . 4  |-  ( (
ph  /\  K  e.  NN0 )  ->  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  =/=  (/) )
29 infmssuzcl 10559 . . . 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 } )
3015, 28, 29sylancr 645 . . 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 }
)
3112, 30eqeltrd 2510 . 2  |-  ( (
ph  /\  K  e.  NN0 )  ->  ( N `  K )  e.  {
n  e.  NN  | 
( ( B `  K )  x.  n
)  e.  ZZ }
)
32 oveq2 6089 . . . 4  |-  ( n  =  ( N `  K )  ->  (
( B `  K
)  x.  n )  =  ( ( B `
 K )  x.  ( N `  K
) ) )
3332eleq1d 2502 . . 3  |-  ( n  =  ( N `  K )  ->  (
( ( B `  K )  x.  n
)  e.  ZZ  <->  ( ( B `  K )  x.  ( N `  K
) )  e.  ZZ ) )
3433elrab 3092 . 2  |-  ( ( N `  K )  e.  { n  e.  NN  |  ( ( B `  K )  x.  n )  e.  ZZ }  <->  ( ( N `  K )  e.  NN  /\  ( ( B `  K )  x.  ( N `  K ) )  e.  ZZ ) )
3531, 34sylib 189 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709    \ cdif 3317    C_ wss 3320   (/)c0 3628   {csn 3814    e. cmpt 4266    Or wor 4502   `'ccnv 4877   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120   NNcn 10000   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   QQcq 10574    seq cseq 11323   0 pc0p 19561  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  elqaalem2  20237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480  df-0p 19562  df-ply 20107  df-coe 20109
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