Theorem coe1term

Description: The coefficient function of a monomial. (Contributed by Mario Carneiro, 26-Jul-2014)

	Ref	Expression
Hypothesis	coe1term.1	⊢ 𝐹 = ( 𝑧 ∈ ℂ ↦ ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) )
Assertion	coe1term	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑀 ) = if ( 𝑀 = 𝑁 , 𝐴 , 0 ) )

Proof

Step	Hyp	Ref	Expression
1		coe1term.1	⊢ 𝐹 = ( 𝑧 ∈ ℂ ↦ ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) )
2	1	coe1termlem	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) ∧ ( 𝐴 ≠ 0 → ( deg ‘ 𝐹 ) = 𝑁 ) ) )
3	2	simpld	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( coeff ‘ 𝐹 ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) )
4	3	fveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑀 ) = ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) ‘ 𝑀 ) )
5	4	3adant3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑀 ) = ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) ‘ 𝑀 ) )
6		eqid	⊢ ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) = ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) )
7		eqeq1	⊢ ( 𝑛 = 𝑀 → ( 𝑛 = 𝑁 ↔ 𝑀 = 𝑁 ) )
8	7	ifbid	⊢ ( 𝑛 = 𝑀 → if ( 𝑛 = 𝑁 , 𝐴 , 0 ) = if ( 𝑀 = 𝑁 , 𝐴 , 0 ) )
9		simp3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → 𝑀 ∈ ℕ0 )
10		simp1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → 𝐴 ∈ ℂ )
11		0cn	⊢ 0 ∈ ℂ
12		ifcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → if ( 𝑀 = 𝑁 , 𝐴 , 0 ) ∈ ℂ )
13	10 11 12	sylancl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → if ( 𝑀 = 𝑁 , 𝐴 , 0 ) ∈ ℂ )
14	6 8 9 13	fvmptd3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑛 ∈ ℕ0 ↦ if ( 𝑛 = 𝑁 , 𝐴 , 0 ) ) ‘ 𝑀 ) = if ( 𝑀 = 𝑁 , 𝐴 , 0 ) )
15	5 14	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( coeff ‘ 𝐹 ) ‘ 𝑀 ) = if ( 𝑀 = 𝑁 , 𝐴 , 0 ) )