This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem zlem1lt

Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004)

Ref Expression
Assertion zlem1lt ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 − 1 ) < 𝑁 ) )

Proof

Step Hyp Ref Expression
1 peano2zm ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 − 1 ) ∈ ℤ )
2 zltp1le ⊢ ( ( ( 𝑀 − 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 − 1 ) < 𝑁 ↔ ( ( 𝑀 − 1 ) + 1 ) ≤ 𝑁 ) )
3 1 2 sylan ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 − 1 ) < 𝑁 ↔ ( ( 𝑀 − 1 ) + 1 ) ≤ 𝑁 ) )
4 zcn ⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ )
5 ax-1cn ⊢ 1 ∈ ℂ
6 npcan ⊢ ( ( 𝑀 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑀 − 1 ) + 1 ) = 𝑀 )
7 4 5 6 sylancl ⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 − 1 ) + 1 ) = 𝑀 )
8 7 adantr ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 − 1 ) + 1 ) = 𝑀 )
9 8 breq1d ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( ( 𝑀 − 1 ) + 1 ) ≤ 𝑁 ↔ 𝑀 ≤ 𝑁 ) )
10 3 9 bitr2d ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 − 1 ) < 𝑁 ) )