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 ) |
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 ) |