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 } ) ) )