Metamath Proof Explorer


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 ) = -∞ )