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	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑀 + 𝐴 ) ≤ 𝑁 → 𝑀 < 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		rpgt0	⊢ ( 𝐴 ∈ ℝ+ → 0 < 𝐴 )
2	1	3ad2ant3	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 0 < 𝐴 )
3		rpre	⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ )
4	3	3ad2ant3	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝐴 ∈ ℝ )
5		simp1	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝑀 ∈ ℝ )
6	4 5	ltaddposd	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( 0 < 𝐴 ↔ 𝑀 < ( 𝑀 + 𝐴 ) ) )
7	2 6	mpbid	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝑀 < ( 𝑀 + 𝐴 ) )
8		simpl	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝑀 ∈ ℝ )
9	3	adantl	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝐴 ∈ ℝ )
10	8 9	readdcld	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( 𝑀 + 𝐴 ) ∈ ℝ )
11	10	3adant2	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( 𝑀 + 𝐴 ) ∈ ℝ )
12		simp2	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → 𝑁 ∈ ℝ )
13		ltletr	⊢ ( ( 𝑀 ∈ ℝ ∧ ( 𝑀 + 𝐴 ) ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 𝑀 < ( 𝑀 + 𝐴 ) ∧ ( 𝑀 + 𝐴 ) ≤ 𝑁 ) → 𝑀 < 𝑁 ) )
14	5 11 12 13	syl3anc	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑀 < ( 𝑀 + 𝐴 ) ∧ ( 𝑀 + 𝐴 ) ≤ 𝑁 ) → 𝑀 < 𝑁 ) )
15	7 14	mpand	⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝑀 + 𝐴 ) ≤ 𝑁 → 𝑀 < 𝑁 ) )