Theorem mul01

Description: Multiplication by 0 . Theorem I.6 of Apostol p. 18. (Contributed by NM, 15-May-1999) (Revised by Scott Fenton, 3-Jan-2013)

	Ref	Expression
Assertion	mul01	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 0 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		mulcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 0 ∈ ℂ ) → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 0 ) = ( 0 · 𝐴 ) )
4		mul02	⊢ ( 𝐴 ∈ ℂ → ( 0 · 𝐴 ) = 0 )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 0 ) = 0 )