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Theorem coefv0 21730
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 9393 . . 3  |-  0  e.  CC
2 coefv0.1 . . . 4  |-  A  =  (coeff `  F )
3 eqid 2443 . . . 4  |-  (deg `  F )  =  (deg
`  F )
42, 3coeid2 21722 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  CC )  ->  ( F `  0 )  =  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
51, 4mpan2 671 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  = 
sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
6 dgrcl 21716 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
7 nn0uz 10910 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
86, 7syl6eleq 2533 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  ( ZZ>= ` 
0 ) )
9 fzss2 11513 . . . 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 11477 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
12 fveq2 5706 . . . . . . 7  |-  ( k  =  0  ->  ( A `  k )  =  ( A ` 
0 ) )
13 oveq2 6114 . . . . . . . 8  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
14 0exp0e1 11885 . . . . . . . 8  |-  ( 0 ^ 0 )  =  1
1513, 14syl6eq 2491 . . . . . . 7  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
1612, 15oveq12d 6124 . . . . . 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 21715 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
19 0nn0 10609 . . . . . . 7  |-  0  e.  NN0
20 ffvelrn 5856 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  0  e.  NN0 )  -> 
( A `  0
)  e.  CC )
2118, 19, 20sylancl 662 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( A `  0 )  e.  CC )
2221mulid1d 9418 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  =  ( A `  0 ) )
2317, 22sylan9eqr 2497 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( A ` 
0 ) )
2421adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  ( A `  0 )  e.  CC )
2523, 24eqeltrd 2517 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  e.  CC )
26 eldifn 3494 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  -.  k  e.  ( 0 ... 0
) )
27 eldifi 3493 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  ( 0 ... (deg `  F ) ) )
28 elfznn0 11496 . . . . . . . . . . . 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 10596 . . . . . . . . . . 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 10672 . . . . . . . . . . 11  |-  0  e.  ZZ
35 elfz3 11476 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
3634, 35ax-mp 5 . . . . . . . . . 10  |-  0  e.  ( 0 ... 0
)
3733, 36syl6eqel 2531 . . . . . . . . 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 466 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  k  e.  NN )
41400expd 12039 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( 0 ^ k )  =  0 )
4241oveq2d 6122 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  ( ( A `  k
)  x.  0 ) )
43 ffvelrn 5856 . . . . . 6  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
4418, 29, 43syl2an 477 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( A `  k )  e.  CC )
4544mul01d 9583 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
4642, 45eqtrd 2475 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  0 )
47 fzfid 11810 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... (deg `  F
) )  e.  Fin )
4810, 25, 46, 47fsumss 13217 . 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 2517 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  e.  CC )
5016fsum1 13233 . . . 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 663 . . 3  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( ( A `  0
)  x.  1 ) )
5251, 22eqtrd 2475 . 2  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( A `  0 ) )
535, 48, 523eqtr2d 2481 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 1369    e. wcel 1756    \ cdif 3340    C_ wss 3343   -->wf 5429   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297   1c1 9298    x. cmul 9302   NNcn 10337   NN0cn0 10594   ZZcz 10661   ZZ>=cuz 10876   ...cfz 11452   ^cexp 11880   sum_csu 13178  Polycply 21667  coeffccoe 21669  degcdgr 21670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-pm 7232  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-sup 7706  df-oi 7739  df-card 8124  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-rlim 12982  df-sum 13179  df-0p 21163  df-ply 21671  df-coe 21673  df-dgr 21674
This theorem is referenced by:  coemulc  21737  dgreq0  21747  vieta1lem2  21792  aareccl  21807  ftalem5  22429  signsply0  26967
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