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Theorem coe1fv 16524
Description: Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
coe1fval.a  |-  A  =  (coe1 `  F )
Assertion
Ref Expression
coe1fv  |-  ( ( F  e.  V  /\  N  e.  NN0 )  -> 
( A `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )

Proof of Theorem coe1fv
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 coe1fval.a . . . 4  |-  A  =  (coe1 `  F )
21coe1fval 16523 . . 3  |-  ( F  e.  V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
32fveq1d 5663 . 2  |-  ( F  e.  V  ->  ( A `  N )  =  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
) )
4 sneq 3761 . . . . 5  |-  ( n  =  N  ->  { n }  =  { N } )
54xpeq2d 4835 . . . 4  |-  ( n  =  N  ->  ( 1o  X.  { n }
)  =  ( 1o 
X.  { N }
) )
65fveq2d 5665 . . 3  |-  ( n  =  N  ->  ( F `  ( 1o  X.  { n } ) )  =  ( F `
 ( 1o  X.  { N } ) ) )
7 eqid 2380 . . 3  |-  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )  =  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )
8 fvex 5675 . . 3  |-  ( F `
 ( 1o  X.  { N } ) )  e.  _V
96, 7, 8fvmpt 5738 . 2  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
103, 9sylan9eq 2432 1  |-  ( ( F  e.  V  /\  N  e.  NN0 )  -> 
( A `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3750    e. cmpt 4200    X. cxp 4809   ` cfv 5387   1oc1o 6646   NN0cn0 10146  coe1cco1 16494
This theorem is referenced by:  fvcoe1  16525  coe1mul2  16582  deg1le0  19894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-i2m1 8984  ax-1ne0 8985  ax-rrecex 8988  ax-cnre 8989
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-recs 6562  df-rdg 6597  df-nn 9926  df-n0 10147  df-coe1 16501
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