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Theorem aalioulem1 20202
Description: Lemma for aaliou 20208. 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 10332 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4 aalioulem1.c . . . . . . 7  |-  ( ph  ->  Y  e.  NN )
54nncnd 9972 . . . . . 6  |-  ( ph  ->  Y  e.  CC )
64nnne0d 10000 . . . . . 6  |-  ( ph  ->  Y  =/=  0 )
73, 5, 6divcld 9746 . . . . 5  |-  ( ph  ->  ( X  /  Y
)  e.  CC )
8 eqid 2404 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2404 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid2 20111 . . . . 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 6055 . . 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 11267 . . . 4  |-  ( ph  ->  ( 0 ... (deg `  F ) )  e. 
Fin )
14 dgrcl 20105 . . . . . 6  |-  ( F  e.  (Poly `  ZZ )  ->  (deg `  F
)  e.  NN0 )
151, 14syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
165, 15expcld 11478 . . . 4  |-  ( ph  ->  ( Y ^ (deg `  F ) )  e.  CC )
17 0z 10249 . . . . . . . 8  |-  0  e.  ZZ
188coef2 20103 . . . . . . . 8  |-  ( ( F  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  F ) : NN0 --> ZZ )
191, 17, 18sylancl 644 . . . . . . 7  |-  ( ph  ->  (coeff `  F ) : NN0 --> ZZ )
20 elfznn0 11039 . . . . . . 7  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  NN0 )
21 ffvelrn 5827 . . . . . . 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 10332 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  CC )
24 expcl 11354 . . . . . 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 9064 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  e.  CC )
2713, 16, 26fsummulc1 12523 . . 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 2436 . 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 11478 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
(deg `  F )
)  e.  CC )
3223, 25, 31mulassd 9067 . . . 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 10332 . . . . . . . . 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 11492 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  =  ( ( X ^ a
)  /  ( Y ^ a ) ) )
3837oveq1d 6055 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) ) )
3934, 36expcld 11478 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  CC )
40 nnexpcl 11349 . . . . . . . . . 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 9972 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  CC )
4341nnne0d 10000 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  =/=  0
)
4439, 42, 31, 43div13d 9770 . . . . . . 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 2436 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
46 elfzelz 11015 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  ZZ )
4746adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  ZZ )
4830nn0zd 10329 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  ZZ )
4929, 35, 47, 48expsubd 11489 . . . . . . . 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 10330 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  ZZ )
52 fznn0sub 11041 . . . . . . . . . 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 11351 . . . . . . . . 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 2479 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( Y ^ (deg `  F
) )  /  ( Y ^ a ) )  e.  ZZ )
57 zexpcl 11351 . . . . . . . 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 10337 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) )  e.  ZZ )
6045, 59eqeltrd 2478 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  e.  ZZ )
6122, 60zmulcld 10337 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) )  e.  ZZ )
6232, 61eqeltrd 2478 . . 3  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
6313, 62fsumzcl 12484 . 2  |-  ( ph  -> 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) )  e.  ZZ )
6428, 63eqeltrd 2478 1  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    x. cmul 8951    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  aalioulem4  20205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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