Theorem ne0p

Description: A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014)

	Ref	Expression
Assertion	ne0p	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐹 ‘ 𝐴 ) ≠ 0 ) → 𝐹 ≠ 0𝑝 )

Proof

Step	Hyp	Ref	Expression
1		0pval	⊢ ( 𝐴 ∈ ℂ → ( 0𝑝 ‘ 𝐴 ) = 0 )
2		fveq1	⊢ ( 𝐹 = 0𝑝 → ( 𝐹 ‘ 𝐴 ) = ( 0𝑝 ‘ 𝐴 ) )
3	2	eqeq1d	⊢ ( 𝐹 = 0𝑝 → ( ( 𝐹 ‘ 𝐴 ) = 0 ↔ ( 0𝑝 ‘ 𝐴 ) = 0 ) )
4	1 3	syl5ibrcom	⊢ ( 𝐴 ∈ ℂ → ( 𝐹 = 0𝑝 → ( 𝐹 ‘ 𝐴 ) = 0 ) )
5	4	necon3d	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐹 ‘ 𝐴 ) ≠ 0 → 𝐹 ≠ 0𝑝 ) )
6	5	imp	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐹 ‘ 𝐴 ) ≠ 0 ) → 𝐹 ≠ 0𝑝 )