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Theorem coefv0 21674
Description: The result of evaluating a polynomial at zero is the constant term. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypothesis
Ref Expression
coefv0.1  |-  A  =  (coeff `  F )
Assertion
Ref Expression
coefv0  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  =  ( A `  0
) )

Proof of Theorem coefv0
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 0cn 9374 . . 3  |-  0  e.  CC
2 coefv0.1 . . . 4  |-  A  =  (coeff `  F )
3 eqid 2441 . . . 4  |-  (deg `  F )  =  (deg
`  F )
42, 3coeid2 21666 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  CC )  ->  ( F `  0 )  =  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
51, 4mpan2 666 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  = 
sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
6 dgrcl 21660 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
7 nn0uz 10891 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
86, 7syl6eleq 2531 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  ( ZZ>= ` 
0 ) )
9 fzss2 11494 . . . 4  |-  ( (deg
`  F )  e.  ( ZZ>= `  0 )  ->  ( 0 ... 0
)  C_  ( 0 ... (deg `  F
) ) )
108, 9syl 16 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... 0 )  C_  ( 0 ... (deg `  F ) ) )
11 elfz1eq 11458 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
12 fveq2 5688 . . . . . . 7  |-  ( k  =  0  ->  ( A `  k )  =  ( A ` 
0 ) )
13 oveq2 6098 . . . . . . . 8  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
14 0exp0e1 11866 . . . . . . . 8  |-  ( 0 ^ 0 )  =  1
1513, 14syl6eq 2489 . . . . . . 7  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
1612, 15oveq12d 6108 . . . . . 6  |-  ( k  =  0  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
1711, 16syl 16 . . . . 5  |-  ( k  e.  ( 0 ... 0 )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
182coef3 21659 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
19 0nn0 10590 . . . . . . 7  |-  0  e.  NN0
20 ffvelrn 5838 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  0  e.  NN0 )  -> 
( A `  0
)  e.  CC )
2118, 19, 20sylancl 657 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( A `  0 )  e.  CC )
2221mulid1d 9399 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  =  ( A `  0 ) )
2317, 22sylan9eqr 2495 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( A ` 
0 ) )
2421adantr 462 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  ( A `  0 )  e.  CC )
2523, 24eqeltrd 2515 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  e.  CC )
26 eldifn 3476 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  -.  k  e.  ( 0 ... 0
) )
27 eldifi 3475 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  ( 0 ... (deg `  F ) ) )
28 elfznn0 11477 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... (deg `  F )
)  ->  k  e.  NN0 )
2927, 28syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  NN0 )
30 elnn0 10577 . . . . . . . . . . 11  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
3129, 30sylib 196 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( k  e.  NN  \/  k  =  0 ) )
3231ord 377 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( -.  k  e.  NN  ->  k  = 
0 ) )
33 id 22 . . . . . . . . . 10  |-  ( k  =  0  ->  k  =  0 )
34 0z 10653 . . . . . . . . . . 11  |-  0  e.  ZZ
35 elfz3 11457 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
3634, 35ax-mp 5 . . . . . . . . . 10  |-  0  e.  ( 0 ... 0
)
3733, 36syl6eqel 2529 . . . . . . . . 9  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
3832, 37syl6 33 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( -.  k  e.  NN  ->  k  e.  ( 0 ... 0
) ) )
3926, 38mt3d 125 . . . . . . 7  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  NN )
4039adantl 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  k  e.  NN )
41400expd 12020 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( 0 ^ k )  =  0 )
4241oveq2d 6106 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  ( ( A `  k
)  x.  0 ) )
43 ffvelrn 5838 . . . . . 6  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
4418, 29, 43syl2an 474 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( A `  k )  e.  CC )
4544mul01d 9564 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
4642, 45eqtrd 2473 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  0 )
47 fzfid 11791 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... (deg `  F
) )  e.  Fin )
4810, 25, 46, 47fsumss 13198 . 2  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  sum_ k  e.  ( 0 ... (deg `  F
) ) ( ( A `  k )  x.  ( 0 ^ k ) ) )
4922, 21eqeltrd 2515 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  e.  CC )
5016fsum1 13214 . . . 4  |-  ( ( 0  e.  ZZ  /\  ( ( A ` 
0 )  x.  1 )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( A `  k )  x.  (
0 ^ k ) )  =  ( ( A `  0 )  x.  1 ) )
5134, 49, 50sylancr 658 . . 3  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( ( A `  0
)  x.  1 ) )
5251, 22eqtrd 2473 . 2  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( A `  0 ) )
535, 48, 523eqtr2d 2479 1  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  =  ( A `  0
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    \ cdif 3322    C_ wss 3325   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283   NNcn 10318   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   ^cexp 11861   sum_csu 13159  Polycply 21611  coeffccoe 21613  degcdgr 21614
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-0p 21107  df-ply 21615  df-coe 21617  df-dgr 21618
This theorem is referenced by:  coemulc  21681  dgreq0  21691  vieta1lem2  21736  aareccl  21751  ftalem5  22373  signsply0  26882
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