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 ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐵𝑆 ≠ ∅ ) ∧ ¬ ( 𝐺𝑆 ) 𝑊 ) → ( 𝐼 ‘ ( 𝐺𝑆 ) ) = 𝑥𝑆 ( 𝐼𝑥 ) )

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 ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐵𝑆 ≠ ∅ ) ∧ ¬ ( 𝐺𝑆 ) 𝑊 ) → ( 𝐼 ‘ ( 𝐺𝑆 ) ) = 𝑥𝑆 ( 𝐼𝑥 ) )