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Metamath Proof Explorer

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 ) )