Theorem mullid

Description: Identity law for multiplication. See mulrid for commuted version. (Contributed by NM, 8-Oct-1999)

	Ref	Expression
Assertion	mullid	⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		mulcom	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 1 · 𝐴 ) = ( 𝐴 · 1 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = ( 𝐴 · 1 ) )
4		mulrid	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )