Theorem dihglbcN

Description: Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		dihglbc.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihglbc.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
3		dihglbc.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dihglbc.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
5		dihglbc.l	⊢ ≤ = ( le ‘ 𝐾 )
6		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
9		eqid	⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
10		eqid	⊢ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
11		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
12		eqid	⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
13		eqid	⊢ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 ) = ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) = 𝑞 )
14	1 2 3 4 5 6 7 8 9 10 11 12 13	dihglbcpreN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ) ∧ ¬ ( 𝐺 ‘ 𝑆 ) ≤ 𝑊 ) → ( 𝐼 ‘ ( 𝐺 ‘ 𝑆 ) ) = ∩ 𝑥 ∈ 𝑆 ( 𝐼 ‘ 𝑥 ) )