Theorem ldil1o

Description: A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013)

	Ref	Expression
Hypotheses	ldil1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ldil1o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ldil1o.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ldil1o	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ldil1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ldil1o.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		ldil1o.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
4		simpll	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐾 ∈ 𝑉 )
5		eqid	⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
6	2 5 3	ldillaut	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
7	1 5	laut1o	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )
8	4 6 7	syl2anc	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝐷 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 )