Metamath Proof Explorer


Theorem xmullid

Description: Extended real version of mullid . (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion xmullid ( 𝐴 ∈ ℝ* → ( 1 ·e 𝐴 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 1xr 1 ∈ ℝ*
2 xmulcom ( ( 1 ∈ ℝ*𝐴 ∈ ℝ* ) → ( 1 ·e 𝐴 ) = ( 𝐴 ·e 1 ) )
3 1 2 mpan ( 𝐴 ∈ ℝ* → ( 1 ·e 𝐴 ) = ( 𝐴 ·e 1 ) )
4 xmulrid ( 𝐴 ∈ ℝ* → ( 𝐴 ·e 1 ) = 𝐴 )
5 3 4 eqtrd ( 𝐴 ∈ ℝ* → ( 1 ·e 𝐴 ) = 𝐴 )