Theorem addlelt

Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021)

	Ref	Expression
Assertion	addlelt	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to \left(M+A\le N\to M<N\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpgt0	$$⊢A\in {ℝ}^{+}\to 0<A$$
2	1	3ad2ant3	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to 0<A$$
3		rpre	$$⊢A\in {ℝ}^{+}\to A\in ℝ$$
4	3	3ad2ant3	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to A\in ℝ$$
5		simp1	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to M\in ℝ$$
6	4 5	ltaddposd	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to \left(0<A\leftrightarrow M<M+A\right)$$
7	2 6	mpbid	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to M<M+A$$
8		simpl	$$⊢\left(M\in ℝ\wedge A\in {ℝ}^{+}\right)\to M\in ℝ$$
9	3	adantl	$$⊢\left(M\in ℝ\wedge A\in {ℝ}^{+}\right)\to A\in ℝ$$
10	8 9	readdcld	$$⊢\left(M\in ℝ\wedge A\in {ℝ}^{+}\right)\to M+A\in ℝ$$
11	10	3adant2	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to M+A\in ℝ$$
12		simp2	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to N\in ℝ$$
13		ltletr	$$⊢\left(M\in ℝ\wedge M+A\in ℝ\wedge N\in ℝ\right)\to \left(\left(M<M+A\wedge M+A\le N\right)\to M<N\right)$$
14	5 11 12 13	syl3anc	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to \left(\left(M<M+A\wedge M+A\le N\right)\to M<N\right)$$
15	7 14	mpand	$$⊢\left(M\in ℝ\wedge N\in ℝ\wedge A\in {ℝ}^{+}\right)\to \left(M+A\le N\to M<N\right)$$