Metamath Proof Explorer

Theorem cdlemg31b0N

Description: TODO: Fix comment. (Contributed by NM, 30-May-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
cdlemg31.n 
|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
Assertion cdlemg31b0N 
|- ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemg31.n 
 |-  N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
9 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> K e. HL )
10 simp2ll 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> P e. A )
11 simp31l 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> v e. A )
12 simp2rl 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> Q e. A )
13 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> W e. H )
14 9 13 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
15 simp2l 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> F e. T )
17 simp33 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P )
18 1 4 5 6 7 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
19 14 15 16 17 18 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
20 simp2r 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( Q e. A /\ -. Q .<_ W ) )
21 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
22 14 16 21 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) .<_ W )
23 19 22 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( ( R ` F ) e. A /\ ( R ` F ) .<_ W ) )
24 simp31 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( v e. A /\ v .<_ W ) )
25 simp32 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> v =/= ( R ` F ) )
26 25 necomd 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) =/= v )
27 1 2 4 5 lhp2atne 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P e. A ) /\ ( ( ( R ` F ) e. A /\ ( R ` F ) .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( R ` F ) =/= v ) -> ( Q .\/ ( R ` F ) ) =/= ( P .\/ v ) )
28 14 20 10 23 24 26 27 syl321anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( Q .\/ ( R ` F ) ) =/= ( P .\/ v ) )
29 28 necomd 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ v ) =/= ( Q .\/ ( R ` F ) ) )
30 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
31 2 3 30 4 2atmat0 
 |-  ( ( ( K e. HL /\ P e. A /\ v e. A ) /\ ( Q e. A /\ ( R ` F ) e. A /\ ( P .\/ v ) =/= ( Q .\/ ( R ` F ) ) ) ) -> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
32 9 10 11 12 19 29 31 syl33anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
33 8 eleq1i 
 |-  ( N e. A <-> ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A )
34 8 eqeq1i 
 |-  ( N = ( 0. ` K ) <-> ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) )
35 33 34 orbi12i 
 |-  ( ( N e. A \/ N = ( 0. ` K ) ) <-> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
36 32 35 sylibr 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ v =/= ( R ` F ) /\ ( F ` P ) =/= P ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )