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Theorem dgreq 20116
Description: If the highest term in a polynomial expression is nonzero, then the polynomial's degree is completely determined. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
dgreq.1  |-  ( ph  ->  F  e.  (Poly `  S ) )
dgreq.2  |-  ( ph  ->  N  e.  NN0 )
dgreq.3  |-  ( ph  ->  A : NN0 --> CC )
dgreq.4  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
dgreq.5  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
dgreq.6  |-  ( ph  ->  ( A `  N
)  =/=  0 )
Assertion
Ref Expression
dgreq  |-  ( ph  ->  (deg `  F )  =  N )
Distinct variable groups:    z, k, A    k, N, z    ph, k,
z
Allowed substitution hints:    S( z, k)    F( z, k)

Proof of Theorem dgreq
StepHypRef Expression
1 dgreq.1 . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 dgreq.2 . . 3  |-  ( ph  ->  N  e.  NN0 )
3 dgreq.3 . . . 4  |-  ( ph  ->  A : NN0 --> CC )
4 elfznn0 11039 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
5 ffvelrn 5827 . . . 4  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
63, 4, 5syl2an 464 . . 3  |-  ( (
ph  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
7 dgreq.5 . . 3  |-  ( ph  ->  F  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( z ^ k
) ) ) )
81, 2, 6, 7dgrle 20115 . 2  |-  ( ph  ->  (deg `  F )  <_  N )
9 dgreq.4 . . . . . 6  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
101, 2, 3, 9, 7coeeq 20099 . . . . 5  |-  ( ph  ->  (coeff `  F )  =  A )
1110fveq1d 5689 . . . 4  |-  ( ph  ->  ( (coeff `  F
) `  N )  =  ( A `  N ) )
12 dgreq.6 . . . 4  |-  ( ph  ->  ( A `  N
)  =/=  0 )
1311, 12eqnetrd 2585 . . 3  |-  ( ph  ->  ( (coeff `  F
) `  N )  =/=  0 )
14 eqid 2404 . . . 4  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2404 . . . 4  |-  (deg `  F )  =  (deg
`  F )
1614, 15dgrub 20106 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  (deg `  F ) )
171, 2, 13, 16syl3anc 1184 . 2  |-  ( ph  ->  N  <_  (deg `  F
) )
18 dgrcl 20105 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
191, 18syl 16 . . . 4  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2019nn0red 10231 . . 3  |-  ( ph  ->  (deg `  F )  e.  RR )
212nn0red 10231 . . 3  |-  ( ph  ->  N  e.  RR )
2220, 21letri3d 9171 . 2  |-  ( ph  ->  ( (deg `  F
)  =  N  <->  ( (deg `  F )  <_  N  /\  N  <_  (deg `  F ) ) ) )
238, 17, 22mpbir2and 889 1  |-  ( ph  ->  (deg `  F )  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2567   {csn 3774   class class class wbr 4172    e. cmpt 4226   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    <_ cle 9077   NN0cn0 10177   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   sum_csu 12434  Polycply 20056  coeffccoe 20058  degcdgr 20059
This theorem is referenced by:  coe1termlem  20129  basellem2  20817
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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