Theorem xmulmnf2

Description: Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xmulmnf2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( -∞ ·e 𝐴 ) = -∞ )

Proof

Step	Hyp	Ref	Expression
1		mnfxr	⊢ -∞ ∈ ℝ*
2		xmulcom	⊢ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( -∞ ·e 𝐴 ) = ( 𝐴 ·e -∞ ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ* → ( -∞ ·e 𝐴 ) = ( 𝐴 ·e -∞ ) )
4	3	adantr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( -∞ ·e 𝐴 ) = ( 𝐴 ·e -∞ ) )
5		xmulmnf1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( 𝐴 ·e -∞ ) = -∞ )
6	4 5	eqtrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( -∞ ·e 𝐴 ) = -∞ )