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Metamath Proof Explorer

Theorem cdlemg31b0a

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

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 cdlemg31b0a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( 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 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> K e. HL )
10 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> P e. A )
11 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> v e. A )
12 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> Q e. A )
13 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
14 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> F e. T )
15 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
16 15 4 5 6 7 trlator0 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( ( R ` F ) e. A \/ ( R ` F ) = ( 0. ` K ) ) )
17 13 14 16 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( ( R ` F ) e. A \/ ( R ` F ) = ( 0. ` K ) ) )
18 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
19 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
20 13 14 19 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( R ` F ) .<_ W )
21 17 20 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( ( ( R ` F ) e. A \/ ( R ` F ) = ( 0. ` K ) ) /\ ( R ` F ) .<_ W ) )
22 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( v e. A /\ v .<_ W ) )
23 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> v =/= ( R ` F ) )
24 23 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( R ` F ) =/= v )
25 1 2 15 4 5 lhp2at0ne 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ P e. A ) /\ ( ( ( ( R ` F ) e. A \/ ( R ` F ) = ( 0. ` K ) ) /\ ( R ` F ) .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( R ` F ) =/= v ) -> ( Q .\/ ( R ` F ) ) =/= ( P .\/ v ) )
26 13 18 10 21 22 24 25 syl321anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( Q .\/ ( R ` F ) ) =/= ( P .\/ v ) )
27 26 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( P .\/ v ) =/= ( Q .\/ ( R ` F ) ) )
28 2 3 15 4 2at0mat0 
 |-  ( ( ( K e. HL /\ P e. A /\ v e. A ) /\ ( Q e. A /\ ( ( R ` F ) e. A \/ ( R ` F ) = ( 0. ` K ) ) /\ ( P .\/ v ) =/= ( Q .\/ ( R ` F ) ) ) ) -> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
29 9 10 11 12 17 27 28 syl33anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
30 8 eleq1i 
 |-  ( N e. A <-> ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A )
31 8 eqeq1i 
 |-  ( N = ( 0. ` K ) <-> ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) )
32 30 31 orbi12i 
 |-  ( ( N e. A \/ N = ( 0. ` K ) ) <-> ( ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) e. A \/ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( 0. ` K ) ) )
33 29 32 sylibr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )