Theorem xmul02

Description: Extended real version of mul02 . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xmul02	⊢ ( 𝐴 ∈ ℝ* → ( 0 ·e 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		xmulcom	⊢ ( ( 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 0 ·e 𝐴 ) = ( 𝐴 ·e 0 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ* → ( 0 ·e 𝐴 ) = ( 𝐴 ·e 0 ) )
4		xmul01	⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ·e 0 ) = 0 )
5	3 4	eqtrd	⊢ ( 𝐴 ∈ ℝ* → ( 0 ·e 𝐴 ) = 0 )