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Theorem bpolylem 28191
Description: Lemma for bpolyval 28192. (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 6098 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x ^ n )  =  ( X ^
n ) )
21oveq1d 6106 . . . . . . . . . 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 3687 . . . . . . . . 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 4379 . . . . . . . 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 2493 . . . . . . 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 27722 . . . . . . 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 2493 . . . . 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 5693 . . . 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 5691 . . . 4  |-  ( m  =  N  ->  ( F `  m )  =  ( F `  N ) )
1311, 12sylan9eqr 2497 . . 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 28190 . . 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 5701 . . 3  |-  ( F `
 N )  e. 
_V
1613, 14, 15ovmpt2a 6221 . 2  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( N BernPoly  X )  =  ( F `  N ) )
17 ltweuz 11784 . . . . 5  |-  <  We  ( ZZ>= `  0 )
18 nn0uz 10895 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
19 weeq2 4709 . . . . . 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 10585 . . . . 5  |-  NN0  e.  _V
23 exse 4684 . . . . 5  |-  ( NN0 
e.  _V  ->  < Se  NN0 )
2422, 23ax-mp 5 . . . 4  |-  < Se  NN0
2521, 24, 9wfr2 27741 . . 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 27663 . . . . . 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 5110 . . . 4  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( F  |`  Pred (  <  ,  NN0 ,  N
) )  =  ( F  |`  ( 0 ... ( N  - 
1 ) ) ) )
3029fveq2d 5695 . . 3  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( G `  ( F  |`  Pred (  <  ,  NN0 ,  N ) ) )  =  ( G `
 ( F  |`  ( 0 ... ( N  -  1 ) ) ) ) )
3121, 24, 9wfr1 27740 . . . . . . 7  |-  F  Fn  NN0
32 fnfun 5508 . . . . . . 7  |-  ( F  Fn  NN0  ->  Fun  F
)
3331, 32ax-mp 5 . . . . . 6  |-  Fun  F
34 ovex 6116 . . . . . 6  |-  ( 0 ... ( N  - 
1 ) )  e. 
_V
35 resfunexg 5943 . . . . . 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 5040 . . . . . . . . . . 11  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  dom  g  =  dom  ( F  |`  ( 0 ... ( N  -  1 ) ) ) )
38 elfznn0 11481 . . . . . . . . . . . . . 14  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  k  e.  NN0 )
3938ssriv 3360 . . . . . . . . . . . . 13  |-  ( 0 ... ( N  - 
1 ) )  C_  NN0
40 fnssres 5524 . . . . . . . . . . . . 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 5510 . . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  dom  g  =  ( 0 ... ( N  - 
1 ) ) )
45 fveq1 5690 . . . . . . . . . . . . 13  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  (
g `  k )  =  ( ( F  |`  ( 0 ... ( N  -  1 ) ) ) `  k
) )
46 fvres 5704 . . . . . . . . . . . . 13  |-  ( k  e.  ( 0 ... ( N  -  1 ) )  ->  (
( F  |`  (
0 ... ( N  - 
1 ) ) ) `
 k )  =  ( F `  k
) )
4745, 46sylan9eq 2495 . . . . . . . . . . . 12  |-  ( ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
g `  k )  =  ( F `  k ) )
4847oveq1d 6106 . . . . . . . . . . 11  |-  ( ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( g `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( F `
 k )  / 
( ( n  -  k )  +  1 ) ) )
4948oveq2d 6107 . . . . . . . . . 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 13184 . . . . . . . . 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 6107 . . . . . . . 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 3687 . . . . . . 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 5695 . . . . . . . 8  |-  ( g  =  ( F  |`  ( 0 ... ( N  -  1 ) ) )  ->  ( # `
 dom  g )  =  ( # `  (
0 ... ( N  - 
1 ) ) ) )
5453csbeq1d 3295 . . . . . . 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 2475 . . . . . 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 6116 . . . . . . 7  |-  ( ( X ^ n )  -  sum_ k  e.  ( 0 ... ( N  -  1 ) ) ( ( n  _C  k )  x.  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) ) ) )  e.  _V
5756csbex 4425 . . . . . 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 5774 . . . . 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 2580 . . . . . . 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 6099 . . . . . . . 8  |-  ( n  =  N  ->  ( X ^ n )  =  ( X ^ N
) )
62 oveq1 6098 . . . . . . . . . 10  |-  ( n  =  N  ->  (
n  _C  k )  =  ( N  _C  k ) )
63 oveq1 6098 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
n  -  k )  =  ( N  -  k ) )
6463oveq1d 6106 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
( n  -  k
)  +  1 )  =  ( ( N  -  k )  +  1 ) )
6564oveq2d 6107 . . . . . . . . . 10  |-  ( n  =  N  ->  (
( F `  k
)  /  ( ( n  -  k )  +  1 ) )  =  ( ( F `
 k )  / 
( ( N  -  k )  +  1 ) ) )
6662, 65oveq12d 6109 . . . . . . . . 9  |-  ( n  =  N  ->  (
( n  _C  k
)  x.  ( ( F `  k )  /  ( ( n  -  k )  +  1 ) ) )  =  ( ( N  _C  k )  x.  ( ( F `  k )  /  (
( N  -  k
)  +  1 ) ) ) )
6766sumeq2sdv 13181 . . . . . . . 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 6109 . . . . . . 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 3312 . . . . . 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 10669 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  N  e.  ZZ )
72 fz01en 11477 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 ... ( N  - 
1 ) )  ~~  ( 1 ... N
) )
7371, 72syl 16 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( 0 ... ( N  - 
1 ) )  ~~  ( 1 ... N
) )
74 fzfi 11794 . . . . . . . . . 10  |-  ( 0 ... ( N  - 
1 ) )  e. 
Fin
75 fzfi 11794 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
Fin
76 hashen 12118 . . . . . . . . . 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 12117 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
8078, 79eqtrd 2475 . . . . . . 7  |-  ( N  e.  NN0  ->  ( # `  ( 0 ... ( N  -  1 ) ) )  =  N )
8180adantr 465 . . . . . 6  |-  ( ( N  e.  NN0  /\  X  e.  CC )  ->  ( # `  (
0 ... ( N  - 
1 ) ) )  =  N )
8281csbeq1d 3295 . . . . 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 5691 . . . . . . . . . . . 12  |-  ( m  =  k  ->  ( F `  m )  =  ( F `  k ) )
8511, 84sylan9eqr 2497 . . . . . . . . . . 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 5701 . . . . . . . . . . 11  |-  ( F `
 k )  e. 
_V
8785, 14, 86ovmpt2a 6221 . . . . . . . . . 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 6106 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  X  e.  CC )  /\  k  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( (
k BernPoly  X )  /  (
( N  -  k
)  +  1 ) )  =  ( ( F `  k )  /  ( ( N  -  k )  +  1 ) ) )
9089oveq2d 6107 . . . . . . 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 13180 . . . . . 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 6107 . . . . 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 2485 . . . 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 2487 . . 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 2475 . 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 2479 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 1369    e. wcel 1756   _Vcvv 2972   [_csb 3288    C_ wss 3328   class class class wbr 4292    e. cmpt 4350   Se wse 4677    We wwe 4678   dom cdm 4840    |` cres 4842   Fun wfun 5412    Fn wfn 5413   ` cfv 5418  (class class class)co 6091    ~~ cen 7307   Fincfn 7310   CCcc 9280   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    - cmin 9595    / cdiv 9993   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437   ^cexp 11865    _C cbc 12078   #chash 12103   sum_csu 13163   Predcpred 27624  wrecscwrecs 27716   BernPoly cbp 28189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807  df-hash 12104  df-sum 13164  df-pred 27625  df-wrecs 27717  df-bpoly 28190
This theorem is referenced by:  bpolyval  28192
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