Theorem ldilset

Description: The set of lattice dilations for a fiducial co-atom W . (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	ldilset	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → 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	ldilfset	⊢ ( 𝐾 ∈ 𝐶 → ( LDil ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) )
7	6	fveq1d	⊢ ( 𝐾 ∈ 𝐶 → ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) ‘ 𝑊 ) )
8		breq2	⊢ ( 𝑤 = 𝑊 → ( 𝑥 ≤ 𝑤 ↔ 𝑥 ≤ 𝑊 ) )
9	8	imbi1d	⊢ ( 𝑤 = 𝑊 → ( ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ) )
10	9	ralbidv	⊢ ( 𝑤 = 𝑊 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ) )
11	10	rabbidv	⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
12		eqid	⊢ ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
13	4	fvexi	⊢ 𝐼 ∈ V
14	13	rabex	⊢ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ∈ V
15	11 12 14	fvmpt	⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑤 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ) ‘ 𝑊 ) = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
16	7 15	sylan9eq	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
17	5 16	syl5eq	⊢ ( ( 𝐾 ∈ 𝐶 ∧ 𝑊 ∈ 𝐻 ) → 𝐷 = { 𝑓 ∈ 𝐼 ∣ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ≤ 𝑊 → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )