Theorem isldil

Description: The predicate "is a lattice dilation". Similar to definition of dilation in Crawley p. 111. (Contributed by NM, 11-May-2012)

	Ref	Expression
Hypotheses	ldilset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ldilset.l	⊢ ≤ = ( le ‘ 𝐾 )
	ldilset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ldilset.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
	ldilset.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
Assertion	isldil	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ldilset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ldilset.l	⊢ ≤ = ( le ‘ 𝐾 )
3		ldilset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ldilset.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
5		ldilset.d	⊢ 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
6	1 2 3 4 5	ldilset	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → 𝐷 = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
7	6	eleq2d	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ 𝐹 ∈ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) )
8		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
9	8	eqeq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) = 𝑥 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑥 ) )
10	9	imbi2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
11	10	ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
12	11	elrab	⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
13	7 12	bitrdi	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 ∈ 𝐷 ↔ ( 𝐹 ∈ 𝐼 ∧ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )