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Metamath Proof Explorer

Theorem lhpmcvr

Description: The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012)

Ref Expression
Hypotheses lhpmcvr.b 
|- B = ( Base ` K )
lhpmcvr.l 
|- .<_ = ( le ` K )
lhpmcvr.m 
|- ./\ = ( meet ` K )
lhpmcvr.c 
|- C = (
lhpmcvr.h 
|- H = ( LHyp ` K )
Assertion lhpmcvr 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) C X )

Proof

Step Hyp Ref Expression
1 lhpmcvr.b 
 |-  B = ( Base ` K )
2 lhpmcvr.l 
 |-  .<_ = ( le ` K )
3 lhpmcvr.m 
 |-  ./\ = ( meet ` K )
4 lhpmcvr.c 
 |-  C = (
5 lhpmcvr.h 
 |-  H = ( LHyp ` K )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> K e. Lat )
8 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> X e. B )
9 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
10 9 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W e. B )
11 1 3 latmcom 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) = ( W ./\ X ) )
12 7 8 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) = ( W ./\ X ) )
13 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
14 13 4 5 lhp1cvr 
 |-  ( ( K e. HL /\ W e. H ) -> W C ( 1. ` K ) )
15 14 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W C ( 1. ` K ) )
16 eqid 
 |-  ( join ` K ) = ( join ` K )
17 1 2 16 13 5 lhpj1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W ( join ` K ) X ) = ( 1. ` K ) )
18 15 17 breqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W C ( W ( join ` K ) X ) )
19 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> K e. HL )
20 1 16 3 4 cvrexch 
 |-  ( ( K e. HL /\ W e. B /\ X e. B ) -> ( ( W ./\ X ) C X <-> W C ( W ( join ` K ) X ) ) )
21 19 10 8 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( W ./\ X ) C X <-> W C ( W ( join ` K ) X ) ) )
22 18 21 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( W ./\ X ) C X )
23 12 22 eqbrtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) C X )