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Theorem aalioulem1 20241
Description: Lemma for aaliou 20247. 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 10368 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4 aalioulem1.c . . . . . . 7  |-  ( ph  ->  Y  e.  NN )
54nncnd 10008 . . . . . 6  |-  ( ph  ->  Y  e.  CC )
64nnne0d 10036 . . . . . 6  |-  ( ph  ->  Y  =/=  0 )
73, 5, 6divcld 9782 . . . . 5  |-  ( ph  ->  ( X  /  Y
)  e.  CC )
8 eqid 2435 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2435 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid2 20150 . . . . 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 6088 . . 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 11304 . . . 4  |-  ( ph  ->  ( 0 ... (deg `  F ) )  e. 
Fin )
14 dgrcl 20144 . . . . . 6  |-  ( F  e.  (Poly `  ZZ )  ->  (deg `  F
)  e.  NN0 )
151, 14syl 16 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
165, 15expcld 11515 . . . 4  |-  ( ph  ->  ( Y ^ (deg `  F ) )  e.  CC )
17 0z 10285 . . . . . . . 8  |-  0  e.  ZZ
188coef2 20142 . . . . . . . 8  |-  ( ( F  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  F ) : NN0 --> ZZ )
191, 17, 18sylancl 644 . . . . . . 7  |-  ( ph  ->  (coeff `  F ) : NN0 --> ZZ )
20 elfznn0 11075 . . . . . . 7  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  NN0 )
21 ffvelrn 5860 . . . . . . 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 10368 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  CC )
24 expcl 11391 . . . . . 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 9100 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  e.  CC )
2713, 16, 26fsummulc1 12560 . . 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 2467 . 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 11515 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
(deg `  F )
)  e.  CC )
3223, 25, 31mulassd 9103 . . . 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 10368 . . . . . . . . 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 11529 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  =  ( ( X ^ a
)  /  ( Y ^ a ) ) )
3837oveq1d 6088 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) ) )
3934, 36expcld 11515 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  CC )
40 nnexpcl 11386 . . . . . . . . . 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 10008 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  CC )
4341nnne0d 10036 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  =/=  0
)
4439, 42, 31, 43div13d 9806 . . . . . . 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 2467 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
46 elfzelz 11051 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  ZZ )
4746adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  ZZ )
4830nn0zd 10365 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  ZZ )
4929, 35, 47, 48expsubd 11526 . . . . . . . 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 10366 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  ZZ )
52 fznn0sub 11077 . . . . . . . . . 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 11388 . . . . . . . . 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 2510 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( Y ^ (deg `  F
) )  /  ( Y ^ a ) )  e.  ZZ )
57 zexpcl 11388 . . . . . . . 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 10373 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) )  e.  ZZ )
6045, 59eqeltrd 2509 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  e.  ZZ )
6122, 60zmulcld 10373 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) )  e.  ZZ )
6232, 61eqeltrd 2509 . . 3  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
6313, 62fsumzcl 12521 . 2  |-  ( ph  -> 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) )  e.  ZZ )
6428, 63eqeltrd 2509 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 2598   -->wf 5442   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   ...cfz 11035   ^cexp 11374   sum_csu 12471  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  aalioulem4  20244
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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