Theorem isdilN

Description: The predicate "is a dilation". (Contributed by NM, 4-Feb-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dilset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dilset.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
	dilset.w	⊢ 𝑊 = ( WAtoms ‘ 𝐾 )
	dilset.m	⊢ 𝑀 = ( PAut ‘ 𝐾 )
	dilset.l	⊢ 𝐿 = ( Dil ‘ 𝐾 )
Assertion	isdilN	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dilset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		dilset.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
3		dilset.w	⊢ 𝑊 = ( WAtoms ‘ 𝐾 )
4		dilset.m	⊢ 𝑀 = ( PAut ‘ 𝐾 )
5		dilset.l	⊢ 𝐿 = ( Dil ‘ 𝐾 )
6	1 2 3 4 5	dilsetN	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐿 ‘ 𝐷 ) = { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } )
7	6	eleq2d	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ 𝐹 ∈ { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 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 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
11	10	ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
12	11	elrab	⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝑀 ∣ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝑓 ‘ 𝑥 ) = 𝑥 ) } ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) )
13	7 12	bitrdi	⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ↔ ( 𝐹 ∈ 𝑀 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝑥 ⊆ ( 𝑊 ‘ 𝐷 ) → ( 𝐹 ‘ 𝑥 ) = 𝑥 ) ) ) )