MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1fv Structured version   Unicode version

Theorem coe1fv 18009
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 18008 . . 3  |-  ( F  e.  V  ->  A  =  ( n  e. 
NN0  |->  ( F `  ( 1o  X.  { n } ) ) ) )
32fveq1d 5859 . 2  |-  ( F  e.  V  ->  ( A `  N )  =  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
) )
4 sneq 4030 . . . . 5  |-  ( n  =  N  ->  { n }  =  { N } )
54xpeq2d 5016 . . . 4  |-  ( n  =  N  ->  ( 1o  X.  { n }
)  =  ( 1o 
X.  { N }
) )
65fveq2d 5861 . . 3  |-  ( n  =  N  ->  ( F `  ( 1o  X.  { n } ) )  =  ( F `
 ( 1o  X.  { N } ) ) )
7 eqid 2460 . . 3  |-  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )  =  ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) )
8 fvex 5867 . . 3  |-  ( F `
 ( 1o  X.  { N } ) )  e.  _V
96, 7, 8fvmpt 5941 . 2  |-  ( N  e.  NN0  ->  ( ( n  e.  NN0  |->  ( F `
 ( 1o  X.  { n } ) ) ) `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
103, 9sylan9eq 2521 1  |-  ( ( F  e.  V  /\  N  e.  NN0 )  -> 
( A `  N
)  =  ( F `
 ( 1o  X.  { N } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {csn 4020    |-> cmpt 4498    X. cxp 4990   ` cfv 5579   1oc1o 7113   NN0cn0 10784  coe1cco1 17981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-om 6672  df-recs 7032  df-rdg 7066  df-nn 10526  df-n0 10785  df-coe1 17986
This theorem is referenced by:  fvcoe1  18010  coe1mul2  18074  deg1le0  22240
  Copyright terms: Public domain W3C validator