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 ) = -∞ ) |
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 ) = -∞ ) |