Theorem ltaddrp2d

Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	ltaddrp2d	$$⊢\phi \to A<B+A$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3	1 2	ltaddrpd	$$⊢\phi \to A<A+B$$
4	1	recnd	$$⊢\phi \to A\in ℂ$$
5	2	rpcnd	$$⊢\phi \to B\in ℂ$$
6	4 5	addcomd	$$⊢\phi \to A+B=B+A$$
7	3 6	breqtrd	$$⊢\phi \to A<B+A$$