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Theorem coe1fv 16287
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 16286 . . 3  |-  ( F  e.  V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
32fveq1d 5527 . 2  |-  ( F  e.  V  ->  ( A `  N )  =  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
) )
4 sneq 3651 . . . . 5  |-  ( n  =  N  ->  { n }  =  { N } )
54xpeq2d 4713 . . . 4  |-  ( n  =  N  ->  ( 1o  X.  { n }
)  =  ( 1o 
X.  { N }
) )
65fveq2d 5529 . . 3  |-  ( n  =  N  ->  ( F `  ( 1o  X.  { n } ) )  =  ( F `
 ( 1o  X.  { N } ) ) )
7 eqid 2283 . . 3  |-  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )  =  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )
8 fvex 5539 . . 3  |-  ( F `
 ( 1o  X.  { N } ) )  e.  _V
96, 7, 8fvmpt 5602 . 2  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
103, 9sylan9eq 2335 1  |-  ( ( F  e.  V  /\  N  e.  NN0 )  -> 
( A `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    X. cxp 4687   ` cfv 5255   1oc1o 6472   NN0cn0 9965  coe1cco1 16255
This theorem is referenced by:  fvcoe1  16288  coe1mul2  16346  deg1le0  19497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-n0 9966  df-coe1 16262
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