Metamath Proof Explorer


Theorem lermy

Description: Y is monotonic (non-strict). (Contributed by Stefan O'Rear, 3-Oct-2014)

Ref Expression
Assertion lermy ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁 ↔ ( 𝐴 Yrm 𝑀 ) ≤ ( 𝐴 Yrm 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 oveq2 ( 𝑎 = 𝑏 → ( 𝐴 Yrm 𝑎 ) = ( 𝐴 Yrm 𝑏 ) )
2 oveq2 ( 𝑎 = 𝑀 → ( 𝐴 Yrm 𝑎 ) = ( 𝐴 Yrm 𝑀 ) )
3 oveq2 ( 𝑎 = 𝑁 → ( 𝐴 Yrm 𝑎 ) = ( 𝐴 Yrm 𝑁 ) )
4 zssre ℤ ⊆ ℝ
5 frmy Yrm : ( ( ℤ ‘ 2 ) × ℤ ) ⟶ ℤ
6 5 fovcl ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑎 ∈ ℤ ) → ( 𝐴 Yrm 𝑎 ) ∈ ℤ )
7 6 zred ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑎 ∈ ℤ ) → ( 𝐴 Yrm 𝑎 ) ∈ ℝ )
8 ltrmy ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) → ( 𝑎 < 𝑏 ↔ ( 𝐴 Yrm 𝑎 ) < ( 𝐴 Yrm 𝑏 ) ) )
9 8 biimpd ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) → ( 𝑎 < 𝑏 → ( 𝐴 Yrm 𝑎 ) < ( 𝐴 Yrm 𝑏 ) ) )
10 9 3expb ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ ( 𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ ) ) → ( 𝑎 < 𝑏 → ( 𝐴 Yrm 𝑎 ) < ( 𝐴 Yrm 𝑏 ) ) )
11 1 2 3 4 7 10 leord1 ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ) → ( 𝑀𝑁 ↔ ( 𝐴 Yrm 𝑀 ) ≤ ( 𝐴 Yrm 𝑁 ) ) )
12 11 3impb ( ( 𝐴 ∈ ( ℤ ‘ 2 ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁 ↔ ( 𝐴 Yrm 𝑀 ) ≤ ( 𝐴 Yrm 𝑁 ) ) )