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Metamath Proof Explorer

Theorem zleltp1

Description: Integer ordering relation. (Contributed by NM, 10-May-2004)

Ref Expression
Assertion zleltp1 ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 zre ⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ )
2 zre ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
3 1re ⊢ 1 ∈ ℝ
4 leadd1 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
5 3 4 mp3an3 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
6 1 2 5 syl2an ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
7 peano2z ⊢ ( 𝑁 ∈ ℤ → ( 𝑁 + 1 ) ∈ ℤ )
8 zltp1le ⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑁 + 1 ) ∈ ℤ ) → ( 𝑀 < ( 𝑁 + 1 ) ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
9 7 8 sylan2 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < ( 𝑁 + 1 ) ↔ ( 𝑀 + 1 ) ≤ ( 𝑁 + 1 ) ) )
10 6 9 bitr4d ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) )