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Metamath Proof Explorer

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 ) )