Theorem xmulpnf2

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

	Ref	Expression
Assertion	xmulpnf2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( +∞ ·e 𝐴 ) = +∞ )

Proof

Step	Hyp	Ref	Expression
1		pnfxr	⊢ +∞ ∈ 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		xmulpnf1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( 𝐴 ·e +∞ ) = +∞ )
6	4 5	eqtrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 < 𝐴 ) → ( +∞ ·e 𝐴 ) = +∞ )