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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 ) = Σ 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 ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) )
4 3 fveq1d ⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐹 ‘ 𝑋 ) = ( ( 𝑧 ∈ ℂ ↦ Σ 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 → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑋 ↑ 𝑘 ) ) )
8 eqid ⊢ ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) )
9 sumex ⊢ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑋 ↑ 𝑘 ) ) ∈ V
10 7 8 9 fvmpt ⊢ ( 𝑋 ∈ ℂ → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑋 ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑋 ↑ 𝑘 ) ) )
11 4 10 sylan9eq ⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑋 ∈ ℂ ) → ( 𝐹 ‘ 𝑋 ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑋 ↑ 𝑘 ) ) )