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Metamath Proof Explorer

Theorem xmul02

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

		Ref	Expression
	Assertion	xmul02	$⊢ A \in ℝ^{*} \to 0 \cdot_{𝑒} A = 0$

Step	Hyp	Ref	Expression
1		0xr	$⊢ 0 \in ℝ^{*}$
2		xmulcom	$⊢ (0 \in ℝ^{} \land A \in ℝ^{}) \to 0 \cdot_{𝑒} A = A \cdot_{𝑒} 0$
3	1 2	mpan	$⊢ A \in ℝ^{*} \to 0 \cdot_{𝑒} A = A \cdot_{𝑒} 0$
4		xmul01	$⊢ A \in ℝ^{*} \to A \cdot_{𝑒} 0 = 0$
5	3 4	eqtrd	$⊢ A \in ℝ^{*} \to 0 \cdot_{𝑒} A = 0$