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Theorem coefv0 20171
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 9089 . . 3  |-  0  e.  CC
2 coefv0.1 . . . 4  |-  A  =  (coeff `  F )
3 eqid 2438 . . . 4  |-  (deg `  F )  =  (deg
`  F )
42, 3coeid2 20163 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  0  e.  CC )  ->  ( F `  0 )  =  sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
51, 4mpan2 654 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  = 
sum_ k  e.  ( 0 ... (deg `  F ) ) ( ( A `  k
)  x.  ( 0 ^ k ) ) )
6 dgrcl 20157 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
7 nn0uz 10525 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
86, 7syl6eleq 2528 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  ( ZZ>= ` 
0 ) )
9 fzss2 11097 . . . 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 11073 . . . . . 6  |-  ( k  e.  ( 0 ... 0 )  ->  k  =  0 )
12 fveq2 5731 . . . . . . 7  |-  ( k  =  0  ->  ( A `  k )  =  ( A ` 
0 ) )
13 oveq2 6092 . . . . . . . 8  |-  ( k  =  0  ->  (
0 ^ k )  =  ( 0 ^ 0 ) )
14 exp0 11391 . . . . . . . . 9  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
151, 14ax-mp 5 . . . . . . . 8  |-  ( 0 ^ 0 )  =  1
1613, 15syl6eq 2486 . . . . . . 7  |-  ( k  =  0  ->  (
0 ^ k )  =  1 )
1712, 16oveq12d 6102 . . . . . 6  |-  ( k  =  0  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
1811, 17syl 16 . . . . 5  |-  ( k  e.  ( 0 ... 0 )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( ( A `
 0 )  x.  1 ) )
192coef3 20156 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
20 0nn0 10241 . . . . . . 7  |-  0  e.  NN0
21 ffvelrn 5871 . . . . . . 7  |-  ( ( A : NN0 --> CC  /\  0  e.  NN0 )  -> 
( A `  0
)  e.  CC )
2219, 20, 21sylancl 645 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  ( A `  0 )  e.  CC )
2322mulid1d 9110 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  =  ( A `  0 ) )
2418, 23sylan9eqr 2492 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  =  ( A ` 
0 ) )
2522adantr 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  ( A `  0 )  e.  CC )
2624, 25eqeltrd 2512 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( 0 ... 0
) )  ->  (
( A `  k
)  x.  ( 0 ^ k ) )  e.  CC )
27 eldifn 3472 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  -.  k  e.  ( 0 ... 0
) )
28 eldifi 3471 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  ( 0 ... (deg `  F ) ) )
29 elfznn0 11088 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... (deg `  F )
)  ->  k  e.  NN0 )
3028, 29syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  NN0 )
31 elnn0 10228 . . . . . . . . . . 11  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
3230, 31sylib 190 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( k  e.  NN  \/  k  =  0 ) )
3332ord 368 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( -.  k  e.  NN  ->  k  = 
0 ) )
34 id 21 . . . . . . . . . 10  |-  ( k  =  0  ->  k  =  0 )
35 0z 10298 . . . . . . . . . . 11  |-  0  e.  ZZ
36 elfz3 11072 . . . . . . . . . . 11  |-  ( 0  e.  ZZ  ->  0  e.  ( 0 ... 0
) )
3735, 36ax-mp 5 . . . . . . . . . 10  |-  0  e.  ( 0 ... 0
)
3834, 37syl6eqel 2526 . . . . . . . . 9  |-  ( k  =  0  ->  k  e.  ( 0 ... 0
) )
3933, 38syl6 32 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  ( -.  k  e.  NN  ->  k  e.  ( 0 ... 0
) ) )
4027, 39mt3d 120 . . . . . . 7  |-  ( k  e.  ( ( 0 ... (deg `  F
) )  \  (
0 ... 0 ) )  ->  k  e.  NN )
4140adantl 454 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  k  e.  NN )
42410expd 11544 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( 0 ^ k )  =  0 )
4342oveq2d 6100 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  ( ( A `  k
)  x.  0 ) )
44 ffvelrn 5871 . . . . . 6  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
4519, 30, 44syl2an 465 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( A `  k )  e.  CC )
4645mul01d 9270 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  0 )  =  0 )
4743, 46eqtrd 2470 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  ( ( 0 ... (deg `  F )
)  \  ( 0 ... 0 ) ) )  ->  ( ( A `  k )  x.  ( 0 ^ k
) )  =  0 )
48 fzfid 11317 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( 0 ... (deg `  F
) )  e.  Fin )
4910, 26, 47, 48fsumss 12524 . 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 ) ) )
5023, 22eqeltrd 2512 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( ( A `  0 )  x.  1 )  e.  CC )
5117fsum1 12540 . . . 4  |-  ( ( 0  e.  ZZ  /\  ( ( A ` 
0 )  x.  1 )  e.  CC )  ->  sum_ k  e.  ( 0 ... 0 ) ( ( A `  k )  x.  (
0 ^ k ) )  =  ( ( A `  0 )  x.  1 ) )
5235, 50, 51sylancr 646 . . 3  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( ( A `  0
)  x.  1 ) )
5352, 23eqtrd 2470 . 2  |-  ( F  e.  (Poly `  S
)  ->  sum_ k  e.  ( 0 ... 0
) ( ( A `
 k )  x.  ( 0 ^ k
) )  =  ( A `  0 ) )
545, 49, 533eqtr2d 2476 1  |-  ( F  e.  (Poly `  S
)  ->  ( F `  0 )  =  ( A `  0
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   -->wf 5453   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995   1c1 8996    x. cmul 9000   NNcn 10005   NN0cn0 10226   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048   ^cexp 11387   sum_csu 12484  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  coemulc  20178  dgreq0  20188  vieta1lem2  20233  aareccl  20248  ftalem5  20864
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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