Theorem lvolset

Description: The set of 3-dim lattice volumes in a Hilbert lattice. (Contributed by NM, 1-Jul-2012)

	Ref	Expression
Hypotheses	lvolset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lvolset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
	lvolset.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
	lvolset.v	⊢ 𝑉 = ( LVols ‘ 𝐾 )
Assertion	lvolset	⊢ ( 𝐾 ∈ 𝐴 → 𝑉 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )

Proof

Step	Hyp	Ref	Expression
1		lvolset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lvolset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3		lvolset.p	⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
4		lvolset.v	⊢ 𝑉 = ( LVols ‘ 𝐾 )
5		elex	⊢ ( 𝐾 ∈ 𝐴 → 𝐾 ∈ V )
6		fveq2	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) )
7	6 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 )
8		fveq2	⊢ ( 𝑘 = 𝐾 → ( LPlanes ‘ 𝑘 ) = ( LPlanes ‘ 𝐾 ) )
9	8 3	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( LPlanes ‘ 𝑘 ) = 𝑃 )
10		fveq2	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = ( ⋖ ‘ 𝐾 ) )
11	10 2	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = 𝐶 )
12	11	breqd	⊢ ( 𝑘 = 𝐾 → ( 𝑦 ( ⋖ ‘ 𝑘 ) 𝑥 ↔ 𝑦 𝐶 𝑥 ) )
13	9 12	rexeqbidv	⊢ ( 𝑘 = 𝐾 → ( ∃ 𝑦 ∈ ( LPlanes ‘ 𝑘 ) 𝑦 ( ⋖ ‘ 𝑘 ) 𝑥 ↔ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 ) )
14	7 13	rabeqbidv	⊢ ( 𝑘 = 𝐾 → { 𝑥 ∈ ( Base ‘ 𝑘 ) ∣ ∃ 𝑦 ∈ ( LPlanes ‘ 𝑘 ) 𝑦 ( ⋖ ‘ 𝑘 ) 𝑥 } = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )
15		df-lvols	⊢ LVols = ( 𝑘 ∈ V ↦ { 𝑥 ∈ ( Base ‘ 𝑘 ) ∣ ∃ 𝑦 ∈ ( LPlanes ‘ 𝑘 ) 𝑦 ( ⋖ ‘ 𝑘 ) 𝑥 } )
16	1	fvexi	⊢ 𝐵 ∈ V
17	16	rabex	⊢ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } ∈ V
18	14 15 17	fvmpt	⊢ ( 𝐾 ∈ V → ( LVols ‘ 𝐾 ) = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )
19	4 18	syl5eq	⊢ ( 𝐾 ∈ V → 𝑉 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )
20	5 19	syl	⊢ ( 𝐾 ∈ 𝐴 → 𝑉 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑦 ∈ 𝑃 𝑦 𝐶 𝑥 } )