Theorem lhpset

Description: The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012)

	Ref	Expression
Hypotheses	lhpset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lhpset.u	⊢ 1 = ( 1. ‘ 𝐾 )
	lhpset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
	lhpset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpset	⊢ ( 𝐾 ∈ 𝐴 → 𝐻 = { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } )

Proof

Step	Hyp	Ref	Expression
1		lhpset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lhpset.u	⊢ 1 = ( 1. ‘ 𝐾 )
3		lhpset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4		lhpset.h	⊢ 𝐻 = ( LHyp ‘ 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		eqidd	⊢ ( 𝑘 = 𝐾 → 𝑤 = 𝑤 )
9		fveq2	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = ( ⋖ ‘ 𝐾 ) )
10	9 3	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( ⋖ ‘ 𝑘 ) = 𝐶 )
11		fveq2	⊢ ( 𝑘 = 𝐾 → ( 1. ‘ 𝑘 ) = ( 1. ‘ 𝐾 ) )
12	11 2	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( 1. ‘ 𝑘 ) = 1 )
13	8 10 12	breq123d	⊢ ( 𝑘 = 𝐾 → ( 𝑤 ( ⋖ ‘ 𝑘 ) ( 1. ‘ 𝑘 ) ↔ 𝑤 𝐶 1 ) )
14	7 13	rabeqbidv	⊢ ( 𝑘 = 𝐾 → { 𝑤 ∈ ( Base ‘ 𝑘 ) ∣ 𝑤 ( ⋖ ‘ 𝑘 ) ( 1. ‘ 𝑘 ) } = { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } )
15		df-lhyp	⊢ LHyp = ( 𝑘 ∈ V ↦ { 𝑤 ∈ ( Base ‘ 𝑘 ) ∣ 𝑤 ( ⋖ ‘ 𝑘 ) ( 1. ‘ 𝑘 ) } )
16	1	fvexi	⊢ 𝐵 ∈ V
17	16	rabex	⊢ { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } ∈ V
18	14 15 17	fvmpt	⊢ ( 𝐾 ∈ V → ( LHyp ‘ 𝐾 ) = { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } )
19	4 18	syl5eq	⊢ ( 𝐾 ∈ V → 𝐻 = { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } )
20	5 19	syl	⊢ ( 𝐾 ∈ 𝐴 → 𝐻 = { 𝑤 ∈ 𝐵 ∣ 𝑤 𝐶 1 } )