Theorem dihglb

Description: Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014)

	Ref	Expression
Hypotheses	dihglb.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihglb.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
	dihglb.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihglb.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihglb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		dihglb.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihglb.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
3		dihglb.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dihglb.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
7		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
8		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
9		eqid	⊢ ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSubSp ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
10		eqid	⊢ ( LSAtoms ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSAtoms ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
11	1 5 6 7 2 3 4 8 9 10	dihglblem6	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) )