Metamath Proof Explorer


Theorem fwddifnval

Description: The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020)

Ref Expression
Hypotheses fwddifnval.1 φ N 0
fwddifnval.2 φ A
fwddifnval.3 φ F : A
fwddifnval.4 φ X
fwddifnval.5 φ k 0 N X + k A
Assertion fwddifnval φ N n F X = k = 0 N ( N k) 1 N k F X + k

Proof

Step Hyp Ref Expression
1 fwddifnval.1 φ N 0
2 fwddifnval.2 φ A
3 fwddifnval.3 φ F : A
4 fwddifnval.4 φ X
5 fwddifnval.5 φ k 0 N X + k A
6 df-fwddifn n = n 0 , f 𝑝𝑚 x y | k 0 n y + k dom f k = 0 n ( n k) 1 n k f x + k
7 6 a1i φ n = n 0 , f 𝑝𝑚 x y | k 0 n y + k dom f k = 0 n ( n k) 1 n k f x + k
8 oveq2 n = N 0 n = 0 N
9 8 adantr n = N f = F 0 n = 0 N
10 dmeq f = F dom f = dom F
11 10 eleq2d f = F y + k dom f y + k dom F
12 11 adantl n = N f = F y + k dom f y + k dom F
13 9 12 raleqbidv n = N f = F k 0 n y + k dom f k 0 N y + k dom F
14 13 rabbidv n = N f = F y | k 0 n y + k dom f = y | k 0 N y + k dom F
15 oveq1 n = N ( n k) = ( N k)
16 15 adantr n = N f = F ( n k) = ( N k)
17 oveq1 n = N n k = N k
18 17 oveq2d n = N 1 n k = 1 N k
19 fveq1 f = F f x + k = F x + k
20 18 19 oveqan12d n = N f = F 1 n k f x + k = 1 N k F x + k
21 16 20 oveq12d n = N f = F ( n k) 1 n k f x + k = ( N k) 1 N k F x + k
22 21 adantr n = N f = F k 0 n ( n k) 1 n k f x + k = ( N k) 1 N k F x + k
23 9 22 sumeq12dv n = N f = F k = 0 n ( n k) 1 n k f x + k = k = 0 N ( N k) 1 N k F x + k
24 14 23 mpteq12dv n = N f = F x y | k 0 n y + k dom f k = 0 n ( n k) 1 n k f x + k = x y | k 0 N y + k dom F k = 0 N ( N k) 1 N k F x + k
25 24 adantl φ n = N f = F x y | k 0 n y + k dom f k = 0 n ( n k) 1 n k f x + k = x y | k 0 N y + k dom F k = 0 N ( N k) 1 N k F x + k
26 cnex V
27 elpm2r V V F : A A F 𝑝𝑚
28 26 26 27 mpanl12 F : A A F 𝑝𝑚
29 3 2 28 syl2anc φ F 𝑝𝑚
30 26 mptrabex x y | k 0 N y + k dom F k = 0 N ( N k) 1 N k F x + k V
31 30 a1i φ x y | k 0 N y + k dom F k = 0 N ( N k) 1 N k F x + k V
32 7 25 1 29 31 ovmpod φ N n F = x y | k 0 N y + k dom F k = 0 N ( N k) 1 N k F x + k
33 fvoveq1 x = X F x + k = F X + k
34 33 oveq2d x = X 1 N k F x + k = 1 N k F X + k
35 34 oveq2d x = X ( N k) 1 N k F x + k = ( N k) 1 N k F X + k
36 35 sumeq2sdv x = X k = 0 N ( N k) 1 N k F x + k = k = 0 N ( N k) 1 N k F X + k
37 36 adantl φ x = X k = 0 N ( N k) 1 N k F x + k = k = 0 N ( N k) 1 N k F X + k
38 3 fdmd φ dom F = A
39 38 adantr φ k 0 N dom F = A
40 5 39 eleqtrrd φ k 0 N X + k dom F
41 40 ralrimiva φ k 0 N X + k dom F
42 oveq1 y = X y + k = X + k
43 42 eleq1d y = X y + k dom F X + k dom F
44 43 ralbidv y = X k 0 N y + k dom F k 0 N X + k dom F
45 44 elrab X y | k 0 N y + k dom F X k 0 N X + k dom F
46 4 41 45 sylanbrc φ X y | k 0 N y + k dom F
47 sumex k = 0 N ( N k) 1 N k F X + k V
48 47 a1i φ k = 0 N ( N k) 1 N k F X + k V
49 32 37 46 48 fvmptd φ N n F X = k = 0 N ( N k) 1 N k F X + k