This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lhpmcvr2

Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013)

Ref Expression
Hypotheses lhpmcvr2.b 
|- B = ( Base ` K )
lhpmcvr2.l 
|- .<_ = ( le ` K )
lhpmcvr2.j 
|- .\/ = ( join ` K )
lhpmcvr2.m 
|- ./\ = ( meet ` K )
lhpmcvr2.a 
|- A = ( Atoms ` K )
lhpmcvr2.h 
|- H = ( LHyp ` K )
Assertion lhpmcvr2 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) )

Proof

Step Hyp Ref Expression
1 lhpmcvr2.b 
 |-  B = ( Base ` K )
2 lhpmcvr2.l 
 |-  .<_ = ( le ` K )
3 lhpmcvr2.j 
 |-  .\/ = ( join ` K )
4 lhpmcvr2.m 
 |-  ./\ = ( meet ` K )
5 lhpmcvr2.a 
 |-  A = ( Atoms ` K )
6 lhpmcvr2.h 
 |-  H = ( LHyp ` K )
7 eqid 
 |-  (
8 1 2 4 7 6 lhpmcvr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) (
9 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> K e. HL )
10 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> X e. B )
11 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
12 11 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W e. B )
13 1 2 3 4 7 5 cvrval5 
 |-  ( ( K e. HL /\ X e. B /\ W e. B ) -> ( ( X ./\ W ) (  E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) ) )
14 9 10 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( X ./\ W ) (  E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) ) )
15 8 14 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) )