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Theorem aalioulem1 20249
Description: Lemma for aaliou 20255. An integer polynomial cannot inflate the denominator of a rational by more than its degree. (Contributed by Stefan O'Rear, 12-Nov-2014.)
Hypotheses
Ref Expression
aalioulem1.a  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
aalioulem1.b  |-  ( ph  ->  X  e.  ZZ )
aalioulem1.c  |-  ( ph  ->  Y  e.  NN )
Assertion
Ref Expression
aalioulem1  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )

Proof of Theorem aalioulem1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 aalioulem1.a . . . . 5  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
2 aalioulem1.b . . . . . . 7  |-  ( ph  ->  X  e.  ZZ )
32zcnd 10376 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4 aalioulem1.c . . . . . . 7  |-  ( ph  ->  Y  e.  NN )
54nncnd 10016 . . . . . 6  |-  ( ph  ->  Y  e.  CC )
64nnne0d 10044 . . . . . 6  |-  ( ph  ->  Y  =/=  0 )
73, 5, 6divcld 9790 . . . . 5  |-  ( ph  ->  ( X  /  Y
)  e.  CC )
8 eqid 2436 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2436 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid2 20158 . . . . 5  |-  ( ( F  e.  (Poly `  ZZ )  /\  ( X  /  Y )  e.  CC )  ->  ( F `  ( X  /  Y ) )  = 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) ) )
111, 7, 10syl2anc 643 . . . 4  |-  ( ph  ->  ( F `  ( X  /  Y ) )  =  sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) ) )
1211oveq1d 6096 . . 3  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  =  ( sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) )  x.  ( Y ^ (deg `  F
) ) ) )
13 fzfid 11312 . . . 4  |-  ( ph  ->  ( 0 ... (deg `  F ) )  e. 
Fin )
14 dgrcl 20152 . . . . . 6  |-  ( F  e.  (Poly `  ZZ )  ->  (deg `  F
)  e.  NN0 )
151, 14syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
165, 15expcld 11523 . . . 4  |-  ( ph  ->  ( Y ^ (deg `  F ) )  e.  CC )
17 0z 10293 . . . . . . . 8  |-  0  e.  ZZ
188coef2 20150 . . . . . . . 8  |-  ( ( F  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  F ) : NN0 --> ZZ )
191, 17, 18sylancl 644 . . . . . . 7  |-  ( ph  ->  (coeff `  F ) : NN0 --> ZZ )
20 elfznn0 11083 . . . . . . 7  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  NN0 )
21 ffvelrn 5868 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ZZ  /\  a  e.  NN0 )  ->  (
(coeff `  F ) `  a )  e.  ZZ )
2219, 20, 21syl2an 464 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  ZZ )
2322zcnd 10376 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  CC )
24 expcl 11399 . . . . . 6  |-  ( ( ( X  /  Y
)  e.  CC  /\  a  e.  NN0 )  -> 
( ( X  /  Y ) ^ a
)  e.  CC )
257, 20, 24syl2an 464 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  e.  CC )
2623, 25mulcld 9108 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  e.  CC )
2713, 16, 26fsummulc1 12568 . . 3  |-  ( ph  ->  ( sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) )  x.  ( Y ^ (deg `  F
) ) )  = 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) ) )
2812, 27eqtrd 2468 . 2  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  = 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) ) )
295adantr 452 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  CC )
3015adantr 452 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  NN0 )
3129, 30expcld 11523 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
(deg `  F )
)  e.  CC )
3223, 25, 31mulassd 9111 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  =  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) ) )
332adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  ZZ )
3433zcnd 10376 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  CC )
356adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  =/=  0
)
3620adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  NN0 )
3734, 29, 35, 36expdivd 11537 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  =  ( ( X ^ a
)  /  ( Y ^ a ) ) )
3837oveq1d 6096 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) ) )
3934, 36expcld 11523 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  CC )
40 nnexpcl 11394 . . . . . . . . . 10  |-  ( ( Y  e.  NN  /\  a  e.  NN0 )  -> 
( Y ^ a
)  e.  NN )
414, 20, 40syl2an 464 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  NN )
4241nncnd 10016 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  CC )
4341nnne0d 10044 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  =/=  0
)
4439, 42, 31, 43div13d 9814 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
4538, 44eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
46 elfzelz 11059 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  ZZ )
4746adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  ZZ )
4830nn0zd 10373 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  ZZ )
4929, 35, 47, 48expsubd 11534 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  =  ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) ) )
504adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  NN )
5150nnzd 10374 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  ZZ )
52 fznn0sub 11085 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  ( (deg `  F )  -  a
)  e.  NN0 )
5352adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (deg `  F )  -  a
)  e.  NN0 )
54 zexpcl 11396 . . . . . . . . 9  |-  ( ( Y  e.  ZZ  /\  ( (deg `  F )  -  a )  e. 
NN0 )  ->  ( Y ^ ( (deg `  F )  -  a
) )  e.  ZZ )
5551, 53, 54syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  e.  ZZ )
5649, 55eqeltrrd 2511 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( Y ^ (deg `  F
) )  /  ( Y ^ a ) )  e.  ZZ )
57 zexpcl 11396 . . . . . . . 8  |-  ( ( X  e.  ZZ  /\  a  e.  NN0 )  -> 
( X ^ a
)  e.  ZZ )
582, 20, 57syl2an 464 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  ZZ )
5956, 58zmulcld 10381 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) )  e.  ZZ )
6045, 59eqeltrd 2510 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  e.  ZZ )
6122, 60zmulcld 10381 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) )  e.  ZZ )
6232, 61eqeltrd 2510 . . 3  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
6313, 62fsumzcl 12529 . 2  |-  ( ph  -> 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) )  e.  ZZ )
6428, 63eqeltrd 2510 1  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990    x. cmul 8995    - cmin 9291    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZcz 10282   ...cfz 11043   ^cexp 11382   sum_csu 12479  Polycply 20103  coeffccoe 20105  degcdgr 20106
This theorem is referenced by:  aalioulem4  20252
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-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  df-dgr 20110
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