Theorem ltaddrp

Description: Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007)

	Ref	Expression
Assertion	ltaddrp	$$⊢\left(A\in ℝ\wedge B\in {ℝ}^{+}\right)\to A<A+B$$

Proof

Step	Hyp	Ref	Expression
1		elrp	$$⊢B\in {ℝ}^{+}\leftrightarrow \left(B\in ℝ\wedge 0<B\right)$$
2		ltaddpos	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(0<B\leftrightarrow A<A+B\right)$$
3	2	biimpd	$$⊢\left(B\in ℝ\wedge A\in ℝ\right)\to \left(0<B\to A<A+B\right)$$
4	3	expcom	$$⊢A\in ℝ\to \left(B\in ℝ\to \left(0<B\to A<A+B\right)\right)$$
5	4	imp32	$$⊢\left(A\in ℝ\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to A<A+B$$
6	1 5	sylan2b	$$⊢\left(A\in ℝ\wedge B\in {ℝ}^{+}\right)\to A<A+B$$