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 ) ) )