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Metamath Proof Explorer

Theorem lhpelim

Description: Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013)

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

Proof

Step Hyp Ref Expression
1 lhpelim.b 
 |-  B = ( Base ` K )
2 lhpelim.l 
 |-  .<_ = ( le ` K )
3 lhpelim.j 
 |-  .\/ = ( join ` K )
4 lhpelim.m 
 |-  ./\ = ( meet ` K )
5 lhpelim.a 
 |-  A = ( Atoms ` K )
6 lhpelim.h 
 |-  H = ( LHyp ` K )
7 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
8 2 4 7 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P ./\ W ) = ( 0. ` K ) )
9 8 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( P ./\ W ) = ( 0. ` K ) )
10 9 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( 0. ` K ) .\/ ( X ./\ W ) ) )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. HL )
12 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> P e. A )
13 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. Lat )
14 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> X e. B )
15 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> W e. H )
16 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
17 15 16 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> W e. B )
18 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
19 13 14 17 18 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( X ./\ W ) e. B )
20 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W )
21 13 14 17 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( X ./\ W ) .<_ W )
22 1 2 3 4 5 atmod4i2 
 |-  ( ( K e. HL /\ ( P e. A /\ ( X ./\ W ) e. B /\ W e. B ) /\ ( X ./\ W ) .<_ W ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( P .\/ ( X ./\ W ) ) ./\ W ) )
23 11 12 19 17 21 22 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P ./\ W ) .\/ ( X ./\ W ) ) = ( ( P .\/ ( X ./\ W ) ) ./\ W ) )
24 hlol 
 |-  ( K e. HL -> K e. OL )
25 11 24 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> K e. OL )
26 1 3 7 olj02 
 |-  ( ( K e. OL /\ ( X ./\ W ) e. B ) -> ( ( 0. ` K ) .\/ ( X ./\ W ) ) = ( X ./\ W ) )
27 25 19 26 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( 0. ` K ) .\/ ( X ./\ W ) ) = ( X ./\ W ) )
28 10 23 27 3eqtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ X e. B ) -> ( ( P .\/ ( X ./\ W ) ) ./\ W ) = ( X ./\ W ) )