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Theorem aalioulem1 19927
Description: Lemma for aaliou 19933. 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 10269 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4 aalioulem1.c . . . . . . 7  |-  ( ph  ->  Y  e.  NN )
54nncnd 9909 . . . . . 6  |-  ( ph  ->  Y  e.  CC )
64nnne0d 9937 . . . . . 6  |-  ( ph  ->  Y  =/=  0 )
73, 5, 6divcld 9683 . . . . 5  |-  ( ph  ->  ( X  /  Y
)  e.  CC )
8 eqid 2366 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2366 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid2 19836 . . . . 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 642 . . . 4  |-  ( ph  ->  ( F `  ( X  /  Y ) )  =  sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) ) )
1211oveq1d 5996 . . 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 11199 . . . 4  |-  ( ph  ->  ( 0 ... (deg `  F ) )  e. 
Fin )
14 dgrcl 19830 . . . . . 6  |-  ( F  e.  (Poly `  ZZ )  ->  (deg `  F
)  e.  NN0 )
151, 14syl 15 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
165, 15expcld 11410 . . . 4  |-  ( ph  ->  ( Y ^ (deg `  F ) )  e.  CC )
17 0z 10186 . . . . . . . 8  |-  0  e.  ZZ
188coef2 19828 . . . . . . . 8  |-  ( ( F  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  F ) : NN0 --> ZZ )
191, 17, 18sylancl 643 . . . . . . 7  |-  ( ph  ->  (coeff `  F ) : NN0 --> ZZ )
20 elfznn0 10975 . . . . . . 7  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  NN0 )
21 ffvelrn 5770 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ZZ  /\  a  e.  NN0 )  ->  (
(coeff `  F ) `  a )  e.  ZZ )
2219, 20, 21syl2an 463 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  ZZ )
2322zcnd 10269 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  CC )
24 expcl 11286 . . . . . 6  |-  ( ( ( X  /  Y
)  e.  CC  /\  a  e.  NN0 )  -> 
( ( X  /  Y ) ^ a
)  e.  CC )
257, 20, 24syl2an 463 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  e.  CC )
2623, 25mulcld 9002 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  e.  CC )
2713, 16, 26fsummulc1 12455 . . 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 2398 . 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 451 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  CC )
3015adantr 451 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  NN0 )
3129, 30expcld 11410 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
(deg `  F )
)  e.  CC )
3223, 25, 31mulassd 9005 . . . 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 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  ZZ )
3433zcnd 10269 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  CC )
356adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  =/=  0
)
3620adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  NN0 )
3734, 29, 35, 36expdivd 11424 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  =  ( ( X ^ a
)  /  ( Y ^ a ) ) )
3837oveq1d 5996 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) ) )
3934, 36expcld 11410 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  CC )
40 nnexpcl 11281 . . . . . . . . . 10  |-  ( ( Y  e.  NN  /\  a  e.  NN0 )  -> 
( Y ^ a
)  e.  NN )
414, 20, 40syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  NN )
4241nncnd 9909 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  CC )
4341nnne0d 9937 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  =/=  0
)
4439, 42, 31, 43div13d 9707 . . . . . . 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 2398 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
46 elfzelz 10951 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  ZZ )
4746adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  ZZ )
4830nn0zd 10266 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  ZZ )
4929, 35, 47, 48expsubd 11421 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  =  ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) ) )
504adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  NN )
5150nnzd 10267 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  ZZ )
52 fznn0sub 10977 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  ( (deg `  F )  -  a
)  e.  NN0 )
5352adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (deg `  F )  -  a
)  e.  NN0 )
54 zexpcl 11283 . . . . . . . . 9  |-  ( ( Y  e.  ZZ  /\  ( (deg `  F )  -  a )  e. 
NN0 )  ->  ( Y ^ ( (deg `  F )  -  a
) )  e.  ZZ )
5551, 53, 54syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  e.  ZZ )
5649, 55eqeltrrd 2441 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( Y ^ (deg `  F
) )  /  ( Y ^ a ) )  e.  ZZ )
57 zexpcl 11283 . . . . . . . 8  |-  ( ( X  e.  ZZ  /\  a  e.  NN0 )  -> 
( X ^ a
)  e.  ZZ )
582, 20, 57syl2an 463 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  ZZ )
5956, 58zmulcld 10274 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) )  e.  ZZ )
6045, 59eqeltrd 2440 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  e.  ZZ )
6122, 60zmulcld 10274 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) )  e.  ZZ )
6232, 61eqeltrd 2440 . . 3  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
6313, 62fsumzcl 12416 . 2  |-  ( ph  -> 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) )  e.  ZZ )
6428, 63eqeltrd 2440 1  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   0cc0 8884    x. cmul 8889    - cmin 9184    / cdiv 9570   NNcn 9893   NN0cn0 10114   ZZcz 10175   ...cfz 10935   ^cexp 11269   sum_csu 12366  Polycply 19781  coeffccoe 19783  degcdgr 19784
This theorem is referenced by:  aalioulem4  19930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-0p 19240  df-ply 19785  df-coe 19787  df-dgr 19788
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