Theorem laut1o

Description: A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012)

	Ref	Expression
Hypotheses	laut1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	laut1o.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	laut1o	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		laut1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		laut1o.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
3		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
4	1 3 2	islaut	⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ( le ‘ 𝐾 ) ( 𝐹 ‘ 𝑦 ) ) ) ) )
5	4	simprbda	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )