Metamath Proof Explorer


Theorem mulid2

Description: Identity law for multiplication. See mulid1 for commuted version. (Contributed by NM, 8-Oct-1999)

Ref Expression
Assertion mulid2 ( 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 mulid1 ( 𝐴 ∈ ℂ → ( 𝐴 · 1 ) = 𝐴 )
5 3 4 eqtrd ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )