Metamath Proof Explorer

Theorem cdlemefrs29pre00

Description: ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat . (Contributed by NM, 29-Mar-2013)

Ref Expression
Hypotheses cdlemefrs29.b 
|- B = ( Base ` K )
cdlemefrs29.l 
|- .<_ = ( le ` K )
cdlemefrs29.j 
|- .\/ = ( join ` K )
cdlemefrs29.m 
|- ./\ = ( meet ` K )
cdlemefrs29.a 
|- A = ( Atoms ` K )
cdlemefrs29.h 
|- H = ( LHyp ` K )
cdlemefrs29.eq 
|- ( s = R -> ( ph <-> ps ) )
Assertion cdlemefrs29pre00 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( ( -. s .<_ W /\ ph ) /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) ) )

Proof

Step Hyp Ref Expression
1 cdlemefrs29.b 
 |-  B = ( Base ` K )
2 cdlemefrs29.l 
 |-  .<_ = ( le ` K )
3 cdlemefrs29.j 
 |-  .\/ = ( join ` K )
4 cdlemefrs29.m 
 |-  ./\ = ( meet ` K )
5 cdlemefrs29.a 
 |-  A = ( Atoms ` K )
6 cdlemefrs29.h 
 |-  H = ( LHyp ` K )
7 cdlemefrs29.eq 
 |-  ( s = R -> ( ph <-> ps ) )
8 anass 
 |-  ( ( ( -. s .<_ W /\ ph ) /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( ph /\ ( s .\/ ( R ./\ W ) ) = R ) ) )
9 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ps )
10 7 pm5.32ri 
 |-  ( ( ph /\ s = R ) <-> ( ps /\ s = R ) )
11 10 baibr 
 |-  ( ps -> ( s = R <-> ( ph /\ s = R ) ) )
12 9 11 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( s = R <-> ( ph /\ s = R ) ) )
13 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
14 2 4 13 5 6 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) )
15 14 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) -> ( R ./\ W ) = ( 0. ` K ) )
16 15 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( R ./\ W ) = ( 0. ` K ) )
17 16 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( s .\/ ( R ./\ W ) ) = ( s .\/ ( 0. ` K ) ) )
18 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> K e. HL )
19 hlol 
 |-  ( K e. HL -> K e. OL )
20 18 19 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> K e. OL )
21 1 5 atbase 
 |-  ( s e. A -> s e. B )
22 21 adantl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> s e. B )
23 1 3 13 olj01 
 |-  ( ( K e. OL /\ s e. B ) -> ( s .\/ ( 0. ` K ) ) = s )
24 20 22 23 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( s .\/ ( 0. ` K ) ) = s )
25 17 24 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( s .\/ ( R ./\ W ) ) = s )
26 25 eqeq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( s .\/ ( R ./\ W ) ) = R <-> s = R ) )
27 26 anbi2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( ph /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( ph /\ s = R ) ) )
28 12 26 27 3bitr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( s .\/ ( R ./\ W ) ) = R <-> ( ph /\ ( s .\/ ( R ./\ W ) ) = R ) ) )
29 28 anbi2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( ph /\ ( s .\/ ( R ./\ W ) ) = R ) ) ) )
30 8 29 bitr4id 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ps ) /\ s e. A ) -> ( ( ( -. s .<_ W /\ ph ) /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) ) )