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Theorem dgreq 20168
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 11088 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
5 ffvelrn 5871 . . . 4  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
63, 4, 5syl2an 465 . . 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 20167 . 2  |-  ( ph  ->  (deg `  F )  <_  N )
9 dgreq.4 . . . . . 6  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
101, 2, 3, 9, 7coeeq 20151 . . . . 5  |-  ( ph  ->  (coeff `  F )  =  A )
1110fveq1d 5733 . . . 4  |-  ( ph  ->  ( (coeff `  F
) `  N )  =  ( A `  N ) )
12 dgreq.6 . . . 4  |-  ( ph  ->  ( A `  N
)  =/=  0 )
1311, 12eqnetrd 2621 . . 3  |-  ( ph  ->  ( (coeff `  F
) `  N )  =/=  0 )
14 eqid 2438 . . . 4  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2438 . . . 4  |-  (deg `  F )  =  (deg
`  F )
1614, 15dgrub 20158 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  (deg `  F ) )
171, 2, 13, 16syl3anc 1185 . 2  |-  ( ph  ->  N  <_  (deg `  F
) )
18 dgrcl 20157 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
191, 18syl 16 . . . 4  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2019nn0red 10280 . . 3  |-  ( ph  ->  (deg `  F )  e.  RR )
212nn0red 10280 . . 3  |-  ( ph  ->  N  e.  RR )
2220, 21letri3d 9220 . 2  |-  ( ph  ->  ( (deg `  F
)  =  N  <->  ( (deg `  F )  <_  N  /\  N  <_  (deg `  F ) ) ) )
238, 17, 22mpbir2and 890 1  |-  ( ph  ->  (deg `  F )  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601   {csn 3816   class class class wbr 4215    e. cmpt 4269   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    <_ cle 9126   NN0cn0 10226   ZZ>=cuz 10493   ...cfz 11048   ^cexp 11387   sum_csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  coe1termlem  20181  basellem2  20869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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