Theorem ldillaut

Description: A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ldillaut.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ldillaut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
	ldillaut.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ldillaut	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 ∈ 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		ldillaut.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		ldillaut.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
3		ldillaut.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6	4 5 1 2 3	isldil	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )
7	6	simprbda	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 ∈ 𝐼 )