Theorem 2timesgt

Description: Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	2timesgt	$$⊢A\in {ℝ}^{+}\to A<2A$$

Proof

Step	Hyp	Ref	Expression
1		rpre	$$⊢A\in {ℝ}^{+}\to A\in ℝ$$
2		id	$$⊢A\in {ℝ}^{+}\to A\in {ℝ}^{+}$$
3	1 2	ltaddrp2d	$$⊢A\in {ℝ}^{+}\to A<A+A$$
4		rpcn	$$⊢A\in {ℝ}^{+}\to A\in ℂ$$
5		2times	$$⊢A\in ℂ\to 2A=A+A$$
6	5	eqcomd	$$⊢A\in ℂ\to A+A=2A$$
7	4 6	syl	$$⊢A\in {ℝ}^{+}\to A+A=2A$$
8	3 7	breqtrd	$$⊢A\in {ℝ}^{+}\to A<2A$$