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Theorem aalioulem1 19675
Description: Lemma for aaliou 19681. 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
StepHypRef Expression
1 aalioulem1.a . . . . 5  |-  ( ph  ->  F  e.  (Poly `  ZZ ) )
2 aalioulem1.b . . . . . . 7  |-  ( ph  ->  X  e.  ZZ )
32zcnd 10086 . . . . . 6  |-  ( ph  ->  X  e.  CC )
4 aalioulem1.c . . . . . . 7  |-  ( ph  ->  Y  e.  NN )
54nncnd 9730 . . . . . 6  |-  ( ph  ->  Y  e.  CC )
64nnne0d 9758 . . . . . 6  |-  ( ph  ->  Y  =/=  0 )
73, 5, 6divcld 9504 . . . . 5  |-  ( ph  ->  ( X  /  Y
)  e.  CC )
8 eqid 2258 . . . . . 6  |-  (coeff `  F )  =  (coeff `  F )
9 eqid 2258 . . . . . 6  |-  (deg `  F )  =  (deg
`  F )
108, 9coeid2 19584 . . . . 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 645 . . . 4  |-  ( ph  ->  ( F `  ( X  /  Y ) )  =  sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  a )  x.  ( ( X  /  Y ) ^ a
) ) )
1211oveq1d 5807 . . 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 11002 . . . 4  |-  ( ph  ->  ( 0 ... (deg `  F ) )  e. 
Fin )
14 dgrcl 19578 . . . . . 6  |-  ( F  e.  (Poly `  ZZ )  ->  (deg `  F
)  e.  NN0 )
151, 14syl 17 . . . . 5  |-  ( ph  ->  (deg `  F )  e.  NN0 )
165, 15expcld 11212 . . . 4  |-  ( ph  ->  ( Y ^ (deg `  F ) )  e.  CC )
17 0z 10003 . . . . . . . 8  |-  0  e.  ZZ
188coef2 19576 . . . . . . . 8  |-  ( ( F  e.  (Poly `  ZZ )  /\  0  e.  ZZ )  ->  (coeff `  F ) : NN0 --> ZZ )
191, 17, 18sylancl 646 . . . . . . 7  |-  ( ph  ->  (coeff `  F ) : NN0 --> ZZ )
20 elfznn0 10789 . . . . . . 7  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  NN0 )
21 ffvelrn 5597 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> ZZ  /\  a  e.  NN0 )  ->  (
(coeff `  F ) `  a )  e.  ZZ )
2219, 20, 21syl2an 465 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  ZZ )
2322zcnd 10086 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  a
)  e.  CC )
24 expcl 11088 . . . . . 6  |-  ( ( ( X  /  Y
)  e.  CC  /\  a  e.  NN0 )  -> 
( ( X  /  Y ) ^ a
)  e.  CC )
257, 20, 24syl2an 465 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  e.  CC )
2623, 25mulcld 8823 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  e.  CC )
2713, 16, 26fsummulc1 12213 . . 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 2290 . 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 453 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  CC )
3015adantr 453 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  NN0 )
3129, 30expcld 11212 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
(deg `  F )
)  e.  CC )
3223, 25, 31mulassd 8826 . . . 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 453 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  ZZ )
3433zcnd 10086 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  X  e.  CC )
356adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  =/=  0
)
3620adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  NN0 )
3734, 29, 35, 36expdivd 11226 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( X  /  Y ) ^
a )  =  ( ( X ^ a
)  /  ( Y ^ a ) ) )
3837oveq1d 5807 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( X ^ a )  /  ( Y ^
a ) )  x.  ( Y ^ (deg `  F ) ) ) )
3934, 36expcld 11212 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  CC )
40 nnexpcl 11083 . . . . . . . . . 10  |-  ( ( Y  e.  NN  /\  a  e.  NN0 )  -> 
( Y ^ a
)  e.  NN )
414, 20, 40syl2an 465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  NN )
4241nncnd 9730 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  e.  CC )
4341nnne0d 9758 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
a )  =/=  0
)
4439, 42, 31, 43div13d 9528 . . . . . . 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 2290 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  =  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) ) )
46 elfzelz 10765 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  a  e.  ZZ )
4746adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  a  e.  ZZ )
4830nn0zd 10083 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  (deg `  F
)  e.  ZZ )
4929, 35, 47, 48expsubd 11223 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  =  ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) ) )
504adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  NN )
5150nnzd 10084 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  Y  e.  ZZ )
52 fznn0sub 10791 . . . . . . . . . 10  |-  ( a  e.  ( 0 ... (deg `  F )
)  ->  ( (deg `  F )  -  a
)  e.  NN0 )
5352adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( (deg `  F )  -  a
)  e.  NN0 )
54 zexpcl 11085 . . . . . . . . 9  |-  ( ( Y  e.  ZZ  /\  ( (deg `  F )  -  a )  e. 
NN0 )  ->  ( Y ^ ( (deg `  F )  -  a
) )  e.  ZZ )
5551, 53, 54syl2anc 645 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( Y ^
( (deg `  F
)  -  a ) )  e.  ZZ )
5649, 55eqeltrrd 2333 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( Y ^ (deg `  F
) )  /  ( Y ^ a ) )  e.  ZZ )
57 zexpcl 11085 . . . . . . . 8  |-  ( ( X  e.  ZZ  /\  a  e.  NN0 )  -> 
( X ^ a
)  e.  ZZ )
582, 20, 57syl2an 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( X ^
a )  e.  ZZ )
5956, 58zmulcld 10091 . . . . . 6  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( Y ^ (deg `  F ) )  / 
( Y ^ a
) )  x.  ( X ^ a ) )  e.  ZZ )
6045, 59eqeltrd 2332 . . . . 5  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( X  /  Y ) ^ a )  x.  ( Y ^ (deg `  F ) ) )  e.  ZZ )
6122, 60zmulcld 10091 . . . 4  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  a
)  x.  ( ( ( X  /  Y
) ^ a )  x.  ( Y ^
(deg `  F )
) ) )  e.  ZZ )
6232, 61eqeltrd 2332 . . 3  |-  ( (
ph  /\  a  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( ( (coeff `  F ) `  a )  x.  (
( X  /  Y
) ^ a ) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
6313, 62fsumzcl 12174 . 2  |-  ( ph  -> 
sum_ a  e.  ( 0 ... (deg `  F ) ) ( ( ( (coeff `  F ) `  a
)  x.  ( ( X  /  Y ) ^ a ) )  x.  ( Y ^
(deg `  F )
) )  e.  ZZ )
6428, 63eqeltrd 2332 1  |-  ( ph  ->  ( ( F `  ( X  /  Y
) )  x.  ( Y ^ (deg `  F
) ) )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   -->wf 4669   ` cfv 4673  (class class class)co 5792   CCcc 8703   0cc0 8705    x. cmul 8710    - cmin 9005    / cdiv 9391   NNcn 9714   NN0cn0 9933   ZZcz 9992   ...cfz 10749   ^cexp 11071   sum_csu 12124  Polycply 19529  coeffccoe 19531  degcdgr 19532
This theorem is referenced by:  aalioulem4  19678
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-fz 10750  df-fzo 10838  df-fl 10892  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-rlim 11929  df-sum 12125  df-0p 18988  df-ply 19533  df-coe 19535  df-dgr 19536
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