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 ( ( ( 𝐾𝑉𝐹𝐼 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )

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 ( 𝐾𝑉 → ( 𝐹𝐼 ↔ ( 𝐹 : 𝐵1-1-onto𝐵 ∧ ∀ 𝑥𝐵𝑦𝐵 ( 𝑥 𝑦 ↔ ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) ) ) )
5 4 simplbda ( ( 𝐾𝑉𝐹𝐼 ) → ∀ 𝑥𝐵𝑦𝐵 ( 𝑥 𝑦 ↔ ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) )
6 breq1 ( 𝑥 = 𝑋 → ( 𝑥 𝑦𝑋 𝑦 ) )
7 fveq2 ( 𝑥 = 𝑋 → ( 𝐹𝑥 ) = ( 𝐹𝑋 ) )
8 7 breq1d ( 𝑥 = 𝑋 → ( ( 𝐹𝑥 ) ( 𝐹𝑦 ) ↔ ( 𝐹𝑋 ) ( 𝐹𝑦 ) ) )
9 6 8 bibi12d ( 𝑥 = 𝑋 → ( ( 𝑥 𝑦 ↔ ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) ↔ ( 𝑋 𝑦 ↔ ( 𝐹𝑋 ) ( 𝐹𝑦 ) ) ) )
10 breq2 ( 𝑦 = 𝑌 → ( 𝑋 𝑦𝑋 𝑌 ) )
11 fveq2 ( 𝑦 = 𝑌 → ( 𝐹𝑦 ) = ( 𝐹𝑌 ) )
12 11 breq2d ( 𝑦 = 𝑌 → ( ( 𝐹𝑋 ) ( 𝐹𝑦 ) ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )
13 10 12 bibi12d ( 𝑦 = 𝑌 → ( ( 𝑋 𝑦 ↔ ( 𝐹𝑋 ) ( 𝐹𝑦 ) ) ↔ ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) ) )
14 9 13 rspc2v ( ( 𝑋𝐵𝑌𝐵 ) → ( ∀ 𝑥𝐵𝑦𝐵 ( 𝑥 𝑦 ↔ ( 𝐹𝑥 ) ( 𝐹𝑦 ) ) → ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) ) )
15 5 14 mpan9 ( ( ( 𝐾𝑉𝐹𝐼 ) ∧ ( 𝑋𝐵𝑌𝐵 ) ) → ( 𝑋 𝑌 ↔ ( 𝐹𝑋 ) ( 𝐹𝑌 ) ) )