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Metamath Proof Explorer

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 ) )