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Metamath Proof Explorer

Theorem lhple

Description: Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014)

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

Proof

Step Hyp Ref Expression
1 lhple.b 
 |-  B = ( Base ` K )
2 lhple.l 
 |-  .<_ = ( le ` K )
3 lhple.j 
 |-  .\/ = ( join ` K )
4 lhple.m 
 |-  ./\ = ( meet ` K )
5 lhple.a 
 |-  A = ( Atoms ` K )
6 lhple.h 
 |-  H = ( LHyp ` K )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. HL )
8 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. Lat )
9 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> P e. A )
10 1 5 atbase 
 |-  ( P e. A -> P e. B )
11 9 10 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> P e. B )
12 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> X e. B )
13 1 3 latjcom 
 |-  ( ( K e. Lat /\ P e. B /\ X e. B ) -> ( P .\/ X ) = ( X .\/ P ) )
14 8 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( P .\/ X ) = ( X .\/ P ) )
15 14 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( P .\/ X ) ./\ W ) = ( ( X .\/ P ) ./\ W ) )
16 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( K e. HL /\ W e. H ) )
17 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> X .<_ W )
18 1 2 3 4 6 lhpmod6i1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ P e. B ) /\ X .<_ W ) -> ( X .\/ ( P ./\ W ) ) = ( ( X .\/ P ) ./\ W ) )
19 16 12 11 17 18 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = ( ( X .\/ P ) ./\ W ) )
20 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
21 2 4 20 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
22 21 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
23 22 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = ( X .\/ ( 0. ` K ) ) )
24 hlol 
 |-  ( K e. HL -> K e. OL )
25 7 24 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> K e. OL )
26 1 3 20 olj01 
 |-  ( ( K e. OL /\ X e. B ) -> ( X .\/ ( 0. ` K ) ) = X )
27 25 12 26 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( 0. ` K ) ) = X )
28 23 27 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( X .\/ ( P ./\ W ) ) = X )
29 15 19 28 3eqtr2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( P .\/ X ) ./\ W ) = X )