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Metamath Proof Explorer

Theorem coe1fv

Description: Value of an evaluated coefficient in a polynomial coefficient vector. (Contributed by Stefan O'Rear, 21-Mar-2015)

Ref Expression
Hypothesis coe1fval.a ⊢ 𝐴 = ( coe1 ‘ 𝐹 )
Assertion coe1fv ( ( 𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑁 ) = ( 𝐹 ‘ ( 1o × { 𝑁 } ) ) )

Proof

Step Hyp Ref Expression
1 coe1fval.a ⊢ 𝐴 = ( coe1 ‘ 𝐹 )
2 1 coe1fval ⊢ ( 𝐹 ∈ 𝑉 → 𝐴 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) )
3 2 fveq1d ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐴 ‘ 𝑁 ) = ( ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) ‘ 𝑁 ) )
4 sneq ⊢ ( 𝑛 = 𝑁 → { 𝑛 } = { 𝑁 } )
5 4 xpeq2d ⊢ ( 𝑛 = 𝑁 → ( 1o × { 𝑛 } ) = ( 1o × { 𝑁 } ) )
6 5 fveq2d ⊢ ( 𝑛 = 𝑁 → ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) = ( 𝐹 ‘ ( 1o × { 𝑁 } ) ) )
7 eqid ⊢ ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) )
8 fvex ⊢ ( 𝐹 ‘ ( 1o × { 𝑁 } ) ) ∈ V
9 6 7 8 fvmpt ⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ0 ↦ ( 𝐹 ‘ ( 1o × { 𝑛 } ) ) ) ‘ 𝑁 ) = ( 𝐹 ‘ ( 1o × { 𝑁 } ) ) )
10 3 9 sylan9eq ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑁 ) = ( 𝐹 ‘ ( 1o × { 𝑁 } ) ) )