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Theorem bpolylem 29403
Description: Lemma for bpolyval 29404. (Contributed by Scott Fenton, 22-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
bpoly.1  |-  G  =  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
bpoly.2  |-  F  = wrecs (  <  ,  NN0 ,  G )
Assertion
Ref Expression
bpolylem  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
Distinct variable groups:    g, k, n, F    g, N, k, n    g, X, k, n
Allowed substitution hints:    G( g, k, n)

Proof of Theorem bpolylem
Dummy variables  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6290 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x ^ n )  =  ( X ^
n ) )
21oveq1d 6298 . . . . . . . . . 10  |-  ( x  =  X  ->  (
( x ^ n
)  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) ) )  =  ( ( X ^ n )  -  sum_ k  e.  dom  g
( ( n  _C  k )  x.  (
( g `  k
)  /  ( ( n  -  k )  +  1 ) ) ) ) )
32csbeq2dv 3835 . . . . . . . . 9  |-  ( x  =  X  ->  [_ ( # `
 dom  g )  /  n ]_ ( ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  = 
[_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
43mpteq2dv 4534 . . . . . . . 8  |-  ( x  =  X  ->  (
g  e.  _V  |->  [_ ( # `  dom  g
)  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )  =  ( g  e.  _V  |->  [_ ( # `  dom  g
)  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) )
5 bpoly.1 . . . . . . . 8  |-  G  =  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
64, 5syl6eqr 2526 . . . . . . 7  |-  ( x  =  X  ->  (
g  e.  _V  |->  [_ ( # `  dom  g
)  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )  =  G )
7 wrecseq3 28934 . . . . . . 7  |-  ( ( g  e.  _V  |->  [_ ( # `  dom  g
)  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )  =  G  -> wrecs (  <  ,  NN0 ,  ( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) )  = wrecs (  <  ,  NN0 ,  G ) )
86, 7syl 16 . . . . . 6  |-  ( x  =  X  -> wrecs (  <  ,  NN0 ,  ( g  e.  _V  |->  [_ ( # `
 dom  g )  /  n ]_ ( ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) ) )  = wrecs (  <  ,  NN0 ,  G ) )
9 bpoly.2 . . . . . 6  |-  F  = wrecs (  <  ,  NN0 ,  G )
108, 9syl6eqr 2526 . . . . 5  |-  ( x  =  X  -> wrecs (  <  ,  NN0 ,  ( g  e.  _V  |->  [_ ( # `
 dom  g )  /  n ]_ ( ( x ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) ) )  =  F )
1110fveq1d 5867 . . . 4  |-  ( x  =  X  ->  (wrecs (  <  ,  NN0 , 
( g  e.  _V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) ) `  m )  =  ( F `  m ) )
12 fveq2 5865 . . . 4  |-  ( m  =  N  ->  ( F `  m )  =  ( F `  N ) )
1311, 12sylan9eqr 2530 . . 3  |-  ( ( m  =  N  /\  x  =  X )  ->  (wrecs (  <  ,  NN0 ,  ( g  e. 
_V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) ) `  m )  =  ( F `  N ) )
14 df-bpoly 29402 . . 3  |- BernPoly  =  ( m  e.  NN0 ,  x  e.  CC  |->  (wrecs (  <  ,  NN0 ,  ( g  e.  _V  |->  [_ ( # `  dom  g
)  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) ) `  m ) )
15 fvex 5875 . . 3  |-  ( F `
 N )  e. 
_V
1613, 14, 15ovmpt2a 6416 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( F `  N ) )
17 ltweuz 12039 . . . . 5  |-  <  We  ( ZZ>= `  0 )
18 nn0uz 11115 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
19 weeq2 4868 . . . . . 6  |-  ( NN0  =  ( ZZ>= `  0
)  ->  (  <  We 
NN0 
<->  <  We  ( ZZ>= ` 
0 ) ) )
2018, 19ax-mp 5 . . . . 5  |-  (  < 
We  NN0  <->  <  We  ( ZZ>= ` 
0 ) )
2117, 20mpbir 209 . . . 4  |-  <  We  NN0
22 nn0ex 10800 . . . . 5  |-  NN0  e.  _V
23 exse 4843 . . . . 5  |-  ( NN0 
e.  _V  ->  < Se  NN0 )
2422, 23ax-mp 5 . . . 4  |-  < Se  NN0
2521, 24, 9wfr2 28953 . . 3  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( G `  ( F  |`  Pred (  <  ,  NN0 ,  N ) ) ) )
2625adantr 465 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( F `  N
)  =  ( G `
 ( F  |`  Pred (  <  ,  NN0 ,  N ) ) ) )
27 prednn0 28875 . . . . . 6  |-  ( N  e.  NN0  ->  Pred (  <  ,  NN0 ,  N
)  =  ( 0 ... ( N  - 
1 ) ) )
2827adantr 465 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  Pred (  <  ,  NN0 ,  N )  =  ( 0 ... ( N  -  1 ) ) )
2928reseq2d 5272 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( F  |`  Pred (  <  ,  NN0 ,  N
) )  =  ( F  |`  ( 0 ... ( N  - 
1 ) ) ) )
3029fveq2d 5869 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( G `  ( F  |`  Pred (  <  ,  NN0 ,  N ) ) )  =  ( G `
 ( F  |`  ( 0 ... ( N  -  1 ) ) ) ) )
3121, 24, 9wfr1 28952 . . . . . . 7  |-  F  Fn  NN0
32 fnfun 5677 . . . . . . 7  |-  ( F  Fn  NN0  ->  Fun  F
)
3331, 32ax-mp 5 . . . . . 6  |-  Fun  F
34 ovex 6308 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  e. 
_V
35 resfunexg 6125 . . . . . 6  |-  ( ( Fun  F  /\  (
0 ... ( N  - 
1 ) )  e. 
_V )  ->  ( F  |`  ( 0 ... ( N  -  1 ) ) )  e. 
_V )
3633, 34, 35mp2an 672 . . . . 5  |-  ( F  |`  ( 0 ... ( N  -  1 ) ) )  e.  _V
37 dmeq 5202 . . . . . . . . . . 11  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  dom  g  =  dom  ( F  |`  ( 0 ... ( N  -  1 ) ) ) )
38 elfznn0 11769 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
3938ssriv 3508 . . . . . . . . . . . . 13  |-  ( 0 ... ( N  - 
1 ) )  C_  NN0
40 fnssres 5693 . . . . . . . . . . . . 13  |-  ( ( F  Fn  NN0  /\  ( 0 ... ( N  -  1 ) )  C_  NN0 )  -> 
( F  |`  (
0 ... ( N  - 
1 ) ) )  Fn  ( 0 ... ( N  -  1 ) ) )
4131, 39, 40mp2an 672 . . . . . . . . . . . 12  |-  ( F  |`  ( 0 ... ( N  -  1 ) ) )  Fn  (
0 ... ( N  - 
1 ) )
42 fndm 5679 . . . . . . . . . . . 12  |-  ( ( F  |`  ( 0 ... ( N  - 
1 ) ) )  Fn  ( 0 ... ( N  -  1 ) )  ->  dom  ( F  |`  ( 0 ... ( N  - 
1 ) ) )  =  ( 0 ... ( N  -  1 ) ) )
4341, 42ax-mp 5 . . . . . . . . . . 11  |-  dom  ( F  |`  ( 0 ... ( N  -  1 ) ) )  =  ( 0 ... ( N  -  1 ) )
4437, 43syl6eq 2524 . . . . . . . . . 10  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  dom  g  =  ( 0 ... ( N  - 
1 ) ) )
45 fveq1 5864 . . . . . . . . . . . . 13  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  (
g `  k )  =  ( ( F  |`  ( 0 ... ( N  -  1 ) ) ) `  k
) )
46 fvres 5879 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  (
( F  |`  (
0 ... ( N  - 
1 ) ) ) `
 k )  =  ( F `  k
) )
4745, 46sylan9eq 2528 . . . . . . . . . . . 12  |-  ( ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
g `  k )  =  ( F `  k ) )
4847oveq1d 6298 . . . . . . . . . . 11  |-  ( ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( g `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( F `
 k )  / 
( ( n  -  k )  +  1 ) ) )
4948oveq2d 6299 . . . . . . . . . 10  |-  ( ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
5044, 49sumeq12rdv 13491 . . . . . . . . 9  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) )  = 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) ) ) )
5150oveq2d 6299 . . . . . . . 8  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  (
( X ^ n
)  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `
 k )  / 
( ( n  -  k )  +  1 ) ) ) )  =  ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
5251csbeq2dv 3835 . . . . . . 7  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  [_ ( # `
 dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  = 
[_ ( # `  dom  g )  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
5344fveq2d 5869 . . . . . . . 8  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  ( # `
 dom  g )  =  ( # `  (
0 ... ( N  - 
1 ) ) ) )
5453csbeq1d 3442 . . . . . . 7  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  [_ ( # `
 dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) ) ) )  =  [_ ( # `  ( 0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) )
5552, 54eqtrd 2508 . . . . . 6  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  [_ ( # `
 dom  g )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  dom  g ( ( n  _C  k )  x.  ( ( g `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  = 
[_ ( # `  (
0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) )
56 ovex 6308 . . . . . . 7  |-  ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) ) ) )  e.  _V
5756csbex 4580 . . . . . 6  |-  [_ ( # `
 ( 0 ... ( N  -  1 ) ) )  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) ) )  e.  _V
5855, 5, 57fvmpt 5949 . . . . 5  |-  ( ( F  |`  ( 0 ... ( N  - 
1 ) ) )  e.  _V  ->  ( G `  ( F  |`  ( 0 ... ( N  -  1 ) ) ) )  = 
[_ ( # `  (
0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) ) )
5936, 58ax-mp 5 . . . 4  |-  ( G `
 ( F  |`  ( 0 ... ( N  -  1 ) ) ) )  = 
[_ ( # `  (
0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) )
60 nfcvd 2630 . . . . . . 7  |-  ( N  e.  NN0  ->  F/_ n
( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) ) ) )
61 oveq2 6291 . . . . . . . 8  |-  ( n  =  N  ->  ( X ^ n )  =  ( X ^ N
) )
62 oveq1 6290 . . . . . . . . . 10  |-  ( n  =  N  ->  (
n  _C  k )  =  ( N  _C  k ) )
63 oveq1 6290 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
6463oveq1d 6298 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
( n  -  k
)  +  1 )  =  ( ( N  -  k )  +  1 ) )
6564oveq2d 6299 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( F `
 k )  / 
( ( N  -  k )  +  1 ) ) )
6662, 65oveq12d 6301 . . . . . . . . 9  |-  ( n  =  N  ->  (
( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( N  _C  k )  x.  ( ( F `  k )  /  (
( N  -  k
)  +  1 ) ) ) )
6766sumeq2sdv 13488 . . . . . . . 8  |-  ( n  =  N  ->  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) )  =  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) ) )
6861, 67oveq12d 6301 . . . . . . 7  |-  ( n  =  N  ->  (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) ) ) )
6960, 68csbiegf 3459 . . . . . 6  |-  ( N  e.  NN0  ->  [_ N  /  n ]_ ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) ) ) )  =  ( ( X ^ N
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  ( ( F `  k )  /  (
( N  -  k
)  +  1 ) ) ) ) )
7069adantr 465 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  [_ N  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( F `  k
)  /  ( ( N  -  k )  +  1 ) ) ) ) )
71 nn0z 10886 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ZZ )
72 fz01en 11712 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 ... ( N  - 
1 ) )  ~~  ( 1 ... N
) )
7371, 72syl 16 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( 0 ... ( N  - 
1 ) )  ~~  ( 1 ... N
) )
74 fzfi 12049 . . . . . . . . . 10  |-  ( 0 ... ( N  - 
1 ) )  e. 
Fin
75 fzfi 12049 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
Fin
76 hashen 12387 . . . . . . . . . 10  |-  ( ( ( 0 ... ( N  -  1 ) )  e.  Fin  /\  ( 1 ... N
)  e.  Fin )  ->  ( ( # `  (
0 ... ( N  - 
1 ) ) )  =  ( # `  (
1 ... N ) )  <-> 
( 0 ... ( N  -  1 ) )  ~~  ( 1 ... N ) ) )
7774, 75, 76mp2an 672 . . . . . . . . 9  |-  ( (
# `  ( 0 ... ( N  -  1 ) ) )  =  ( # `  (
1 ... N ) )  <-> 
( 0 ... ( N  -  1 ) )  ~~  ( 1 ... N ) )
7873, 77sylibr 212 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 0 ... ( N  -  1 ) ) )  =  (
# `  ( 1 ... N ) ) )
79 hashfz1 12386 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
8078, 79eqtrd 2508 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 0 ... ( N  -  1 ) ) )  =  N )
8180adantr 465 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( # `  (
0 ... ( N  - 
1 ) ) )  =  N )
8281csbeq1d 3442 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  [_ ( # `  (
0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  = 
[_ N  /  n ]_ ( ( X ^
n )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) )
83 simpr 461 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  X  e.  CC )
84 fveq2 5865 . . . . . . . . . . . 12  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
8511, 84sylan9eqr 2530 . . . . . . . . . . 11  |-  ( ( m  =  k  /\  x  =  X )  ->  (wrecs (  <  ,  NN0 ,  ( g  e. 
_V  |->  [_ ( # `  dom  g )  /  n ]_ ( ( x ^
n )  -  sum_ k  e.  dom  g ( ( n  _C  k
)  x.  ( ( g `  k )  /  ( ( n  -  k )  +  1 ) ) ) ) ) ) `  m )  =  ( F `  k ) )
86 fvex 5875 . . . . . . . . . . 11  |-  ( F `
 k )  e. 
_V
8785, 14, 86ovmpt2a 6416 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  X  e.  CC )  ->  ( k BernPoly  X )  =  ( F `  k ) )
8838, 83, 87syl2anr 478 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( k BernPoly  X )  =  ( F `
 k ) )
8988oveq1d 6298 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  =  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) )
9089oveq2d 6299 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( N  _C  k )  x.  ( ( k BernPoly  X
)  /  ( ( N  -  k )  +  1 ) ) )  =  ( ( N  _C  k )  x.  ( ( F `
 k )  / 
( ( N  -  k )  +  1 ) ) ) )
9190sumeq2dv 13487 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  -> 
sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) )  =  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( F `  k
)  /  ( ( N  -  k )  +  1 ) ) ) )
9291oveq2d 6299 . . . . 5  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) ) ) )
9370, 82, 923eqtr4d 2518 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  [_ ( # `  (
0 ... ( N  - 
1 ) ) )  /  n ]_ (
( X ^ n
)  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  ( ( F `  k )  /  (
( n  -  k
)  +  1 ) ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
9459, 93syl5eq 2520 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( G `  ( F  |`  ( 0 ... ( N  -  1 ) ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
9530, 94eqtrd 2508 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( G `  ( F  |`  Pred (  <  ,  NN0 ,  N ) ) )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( N  _C  k )  x.  (
( k BernPoly  X )  /  ( ( N  -  k )  +  1 ) ) ) ) )
9616, 26, 953eqtrd 2512 1  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( ( X ^ N )  -  sum_ k  e.  ( 0 ... ( N  - 
1 ) ) ( ( N  _C  k
)  x.  ( ( k BernPoly  X )  /  (
( N  -  k
)  +  1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   Se wse 4836    We wwe 4837   dom cdm 4999    |` cres 5001   Fun wfun 5581    Fn wfn 5582   ` cfv 5587  (class class class)co 6283    ~~ cen 7513   Fincfn 7516   CCcc 9489   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    < clt 9627    - cmin 9804    / cdiv 10205   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671   ^cexp 12133    _C cbc 12347   #chash 12372   sum_csu 13470   Predcpred 28836  wrecscwrecs 28928   BernPoly cbp 29401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-seq 12075  df-hash 12373  df-sum 13471  df-pred 28837  df-wrecs 28929  df-bpoly 29402
This theorem is referenced by:  bpolyval  29404
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