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Metamath Proof Explorer

Theorem lautle

Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012)

Ref Expression
Hypotheses lautset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lautset.l ⊢ ≤ = ( le ‘ 𝐾 )
lautset.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion lautle ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 lautset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 lautset.l ⊢ ≤ = ( le ‘ 𝐾 )
3 lautset.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
4 1 2 3 islaut ⊢ ( 𝐾 ∈ 𝑉 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )
5 4 simplbda ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
6 breq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦 ) )
7 fveq2 ⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) )
8 7 breq1d ⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
9 6 8 bibi12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
10 breq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌 ) )
11 fveq2 ⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑌 ) )
12 11 breq2d ⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )
13 10 12 bibi12d ⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) )
14 9 13 rspc2v ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) )
15 5 14 mpan9 ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) )