Theorem xmulpnf2

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

	Ref	Expression
Assertion	xmulpnf2	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{+\infty }{\cdot }_{𝑒}A=\mathrm{+\infty }$$

Proof

Step	Hyp	Ref	Expression
1		pnfxr	$$⊢\mathrm{+\infty }\in {ℝ}^{*}$$
2		xmulcom	$$⊢\left(\mathrm{+\infty }\in {ℝ}^{*}\wedge A\in {ℝ}^{*}\right)\to \mathrm{+\infty }{\cdot }_{𝑒}A=A{\cdot }_{𝑒}\mathrm{+\infty }$$
3	1 2	mpan	$$⊢A\in {ℝ}^{*}\to \mathrm{+\infty }{\cdot }_{𝑒}A=A{\cdot }_{𝑒}\mathrm{+\infty }$$
4	3	adantr	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{+\infty }{\cdot }_{𝑒}A=A{\cdot }_{𝑒}\mathrm{+\infty }$$
5		xmulpnf1	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to A{\cdot }_{𝑒}\mathrm{+\infty }=\mathrm{+\infty }$$
6	4 5	eqtrd	$$⊢\left(A\in {ℝ}^{*}\wedge 0<A\right)\to \mathrm{+\infty }{\cdot }_{𝑒}A=\mathrm{+\infty }$$