Theorem idlaut

Description: The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012)

	Ref	Expression
Hypotheses	idlaut.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	idlaut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	idlaut	⊢ ( 𝐾 ∈ 𝐴 → ( I ↾ 𝐵 ) ∈ 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		idlaut.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		idlaut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
3		f1oi	⊢ ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵
4	3	a1i	⊢ ( 𝐾 ∈ 𝐴 → ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵 )
5		fvresi	⊢ ( 𝑥 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑥 ) = 𝑥 )
6		fvresi	⊢ ( 𝑦 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑦 ) = 𝑦 )
7	5 6	breqan12d	⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( I ↾ 𝐵 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑦 ) ↔ 𝑥 ( le ‘ 𝐾 ) 𝑦 ) )
8	7	bicomd	⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( I ↾ 𝐵 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑦 ) ) )
9	8	rgen2	⊢ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( I ↾ 𝐵 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑦 ) )
10	9	a1i	⊢ ( 𝐾 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( I ↾ 𝐵 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑦 ) ) )
11		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12	1 11 2	islaut	⊢ ( 𝐾 ∈ 𝐴 → ( ( I ↾ 𝐵 ) ∈ 𝐼 ↔ ( ( I ↾ 𝐵 ) : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( le ‘ 𝐾 ) 𝑦 ↔ ( ( I ↾ 𝐵 ) ‘ 𝑥 ) ( le ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑦 ) ) ) ) )
13	4 10 12	mpbir2and	⊢ ( 𝐾 ∈ 𝐴 → ( I ↾ 𝐵 ) ∈ 𝐼 )