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Theorem dgreq 22371
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 11761 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
5 ffvelrn 6012 . . . 4  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
63, 4, 5syl2an 477 . . 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 22370 . 2  |-  ( ph  ->  (deg `  F )  <_  N )
9 dgreq.4 . . . . . 6  |-  ( ph  ->  ( A " ( ZZ>=
`  ( N  + 
1 ) ) )  =  { 0 } )
101, 2, 3, 9, 7coeeq 22354 . . . . 5  |-  ( ph  ->  (coeff `  F )  =  A )
1110fveq1d 5861 . . . 4  |-  ( ph  ->  ( (coeff `  F
) `  N )  =  ( A `  N ) )
12 dgreq.6 . . . 4  |-  ( ph  ->  ( A `  N
)  =/=  0 )
1311, 12eqnetrd 2755 . . 3  |-  ( ph  ->  ( (coeff `  F
) `  N )  =/=  0 )
14 eqid 2462 . . . 4  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2462 . . . 4  |-  (deg `  F )  =  (deg
`  F )
1614, 15dgrub 22361 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  (deg `  F ) )
171, 2, 13, 16syl3anc 1223 . 2  |-  ( ph  ->  N  <_  (deg `  F
) )
18 dgrcl 22360 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
191, 18syl 16 . . . 4  |-  ( ph  ->  (deg `  F )  e.  NN0 )
2019nn0red 10844 . . 3  |-  ( ph  ->  (deg `  F )  e.  RR )
212nn0red 10844 . . 3  |-  ( ph  ->  N  e.  RR )
2220, 21letri3d 9717 . 2  |-  ( ph  ->  ( (deg `  F
)  =  N  <->  ( (deg `  F )  <_  N  /\  N  <_  (deg `  F ) ) ) )
238, 17, 22mpbir2and 915 1  |-  ( ph  ->  (deg `  F )  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    =/= wne 2657   {csn 4022   class class class wbr 4442    |-> cmpt 4500   "cima 4997   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    <_ cle 9620   NN0cn0 10786   ZZ>=cuz 11073   ...cfz 11663   ^cexp 12124   sum_csu 13459  Polycply 22311  coeffccoe 22313  degcdgr 22314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-n0 10787  df-z 10856  df-uz 11074  df-rp 11212  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-hash 12363  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-sum 13460  df-0p 21807  df-ply 22315  df-coe 22317  df-dgr 22318
This theorem is referenced by:  coe1termlem  22384  basellem2  23078
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