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Metamath Proof Explorer

Theorem coeid2

Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014)

Ref Expression
Hypotheses dgrub.1 ⊢ A = coeff ⁡ F
dgrub.2 ⊢ N = deg ⁡ F
Assertion coeid2 ⊢ F ∈ Poly ⁡ S ∧ X ∈ ℂ → F ⁡ X = ∑ k = 0 N A ⁡ k ⁢ X k

Proof

Step Hyp Ref Expression
1 dgrub.1 ⊢ A = coeff ⁡ F
2 dgrub.2 ⊢ N = deg ⁡ F
3 1 2 coeid ⊢ F ∈ Poly ⁡ S → F = z ∈ ℂ ⟼ ∑ k = 0 N A ⁡ k ⁢ z k
4 3 fveq1d ⊢ F ∈ Poly ⁡ S → F ⁡ X = z ∈ ℂ ⟼ ∑ k = 0 N A ⁡ k ⁢ z k ⁡ X
5 oveq1 ⊢ z = X → z k = X k
6 5 oveq2d ⊢ z = X → A ⁡ k ⁢ z k = A ⁡ k ⁢ X k
7 6 sumeq2sdv ⊢ z = X → ∑ k = 0 N A ⁡ k ⁢ z k = ∑ k = 0 N A ⁡ k ⁢ X k
8 eqid ⊢ z ∈ ℂ ⟼ ∑ k = 0 N A ⁡ k ⁢ z k = z ∈ ℂ ⟼ ∑ k = 0 N A ⁡ k ⁢ z k
9 sumex ⊢ ∑ k = 0 N A ⁡ k ⁢ X k ∈ V
10 7 8 9 fvmpt ⊢ X ∈ ℂ → z ∈ ℂ ⟼ ∑ k = 0 N A ⁡ k ⁢ z k ⁡ X = ∑ k = 0 N A ⁡ k ⁢ X k
11 4 10 sylan9eq ⊢ F ∈ Poly ⁡ S ∧ X ∈ ℂ → F ⁡ X = ∑ k = 0 N A ⁡ k ⁢ X k