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Metamath Proof Explorer

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 ) )