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Theorem coefv0 22829
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 9538 . . 3  |-  0  e.  CC
2 coefv0.1 . . . 4  |-  A  =  (coeff `  F )
3 eqid 2402 . . . 4  |-  (deg `  F )  =  (deg
`  F )
42, 3coeid2 22820 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  CC )  ->  ( F `  0 )  =  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
51, 4mpan2 669 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  = 
sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
6 dgrcl 22814 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
7 nn0uz 11079 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
86, 7syl6eleq 2500 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  ( ZZ>= ` 
0 ) )
9 fzss2 11695 . . . 4  |-  ( (deg
`  F )  e.  ( ZZ>= `  0 )  ->  ( 0 ... 0
)  C_  ( 0 ... (deg `  F
) ) )
108, 9syl 17 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... 0 )  C_  ( 0 ... (deg `  F ) ) )
11 elfz1eq 11668 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
12 fveq2 5805 . . . . . . 7  |-  ( k  =  0  ->  ( A `  k )  =  ( A ` 
0 ) )
13 oveq2 6242 . . . . . . . 8  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
14 0exp0e1 12125 . . . . . . . 8  |-  ( 0 ^ 0 )  =  1
1513, 14syl6eq 2459 . . . . . . 7  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
1612, 15oveq12d 6252 . . . . . 6  |-  ( k  =  0  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
1711, 16syl 17 . . . . 5  |-  ( k  e.  ( 0 ... 0 )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
182coef3 22813 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
19 0nn0 10771 . . . . . . 7  |-  0  e.  NN0
20 ffvelrn 5963 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  0  e.  NN0 )  -> 
( A `  0
)  e.  CC )
2118, 19, 20sylancl 660 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( A `  0 )  e.  CC )
2221mulid1d 9563 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  =  ( A `  0 ) )
2317, 22sylan9eqr 2465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( A ` 
0 ) )
2421adantr 463 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  ( A `  0 )  e.  CC )
2523, 24eqeltrd 2490 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  e.  CC )
26 eldifn 3565 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  -.  k  e.  ( 0 ... 0
) )
27 eldifi 3564 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  ( 0 ... (deg `  F ) ) )
28 elfznn0 11743 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... (deg `  F )
)  ->  k  e.  NN0 )
2927, 28syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  NN0 )
30 elnn0 10758 . . . . . . . . . . 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 375 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( -.  k  e.  NN  ->  k  = 
0 ) )
33 id 22 . . . . . . . . . 10  |-  ( k  =  0  ->  k  =  0 )
34 0z 10836 . . . . . . . . . . 11  |-  0  e.  ZZ
35 elfz3 11667 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
3634, 35ax-mp 5 . . . . . . . . . 10  |-  0  e.  ( 0 ... 0
)
3733, 36syl6eqel 2498 . . . . . . . . 9  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
3832, 37syl6 31 . . . . . . . 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 464 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  k  e.  NN )
41400expd 12280 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( 0 ^ k )  =  0 )
4241oveq2d 6250 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  ( ( A `  k
)  x.  0 ) )
43 ffvelrn 5963 . . . . . 6  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
4418, 29, 43syl2an 475 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( A `  k )  e.  CC )
4544mul01d 9733 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
4642, 45eqtrd 2443 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  0 )
47 fzfid 12037 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... (deg `  F
) )  e.  Fin )
4810, 25, 46, 47fsumss 13603 . 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 2490 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  e.  CC )
5016fsum1 13620 . . . 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 661 . . 3  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( ( A `  0
)  x.  1 ) )
5251, 22eqtrd 2443 . 2  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( A `  0 ) )
535, 48, 523eqtr2d 2449 1  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  =  ( A `  0
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    \ cdif 3410    C_ wss 3413   -->wf 5521   ` cfv 5525  (class class class)co 6234   CCcc 9440   0cc0 9442   1c1 9443    x. cmul 9447   NNcn 10496   NN0cn0 10756   ZZcz 10825   ZZ>=cuz 11045   ...cfz 11643   ^cexp 12120   sum_csu 13564  Polycply 22765  coeffccoe 22767  degcdgr 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520  ax-addf 9521
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-of 6477  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-pm 7380  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-rlim 13368  df-sum 13565  df-0p 22261  df-ply 22769  df-coe 22771  df-dgr 22772
This theorem is referenced by:  coemulc  22836  dgreq0  22846  vieta1lem2  22891  aareccl  22906  ftalem5  23623  signsply0  28894  elaa2  37367
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