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Metamath Proof Explorer

Theorem cdlemg31c

Description: Show that when N is an atom, it is not under W . TODO: Is there a shorter direct proof? TODO: should we eliminate ( FP ) =/= P here? (Contributed by NM, 29-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 cdlemg31c 
|- ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> -. N .<_ W )

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 simp11l 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> K e. HL )
10 simp11r 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> W e. H )
11 9 10 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 ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( K e. HL /\ W e. H ) )
12 simp13 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> ( Q e. A /\ -. Q .<_ W ) )
13 simp31 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> v =/= ( R ` F ) )
14 13 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 ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) =/= v )
15 simp12 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simp2r 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> F e. T )
17 simp32 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> ( 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 11 15 16 17 18 syl112anc 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) e. A )
20 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
21 11 16 20 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 ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> ( R ` F ) .<_ W )
22 simp2l 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> ( v e. A /\ v .<_ W ) )
23 1 2 4 5 lhp2atnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R ` F ) =/= v ) /\ ( ( R ` F ) e. A /\ ( R ` F ) .<_ W ) /\ ( v e. A /\ v .<_ W ) ) -> -. v .<_ ( Q .\/ ( R ` F ) ) )
24 11 12 14 19 21 22 23 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 ) /\ ( F ` P ) =/= P /\ N e. A ) ) -> -. v .<_ ( Q .\/ ( R ` F ) ) )
25 simp12l 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> P e. A )
26 simp13l 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> Q e. A )
27 simp2ll 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> v e. A )
28 1 2 3 4 5 6 7 8 cdlemg31a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( v e. A /\ F e. T ) ) -> N .<_ ( P .\/ v ) )
29 9 10 25 26 27 16 28 syl222anc 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> N .<_ ( P .\/ v ) )
30 29 adantr 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N .<_ ( P .\/ v ) )
31 simp111 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( K e. HL /\ W e. H ) )
32 simp112 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( P e. A /\ -. P .<_ W ) )
33 simp3 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N =/= v )
34 33 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 ) /\ ( F ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> v =/= N )
35 simp12l 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> ( v e. A /\ v .<_ W ) )
36 simp133 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N e. A )
37 simp2 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> N .<_ W )
38 1 2 4 5 lhp2atnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ v =/= N ) /\ ( v e. A /\ v .<_ W ) /\ ( N e. A /\ N .<_ W ) ) -> -. N .<_ ( P .\/ v ) )
39 31 32 34 35 36 37 38 syl312anc 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W /\ N =/= v ) -> -. N .<_ ( P .\/ v ) )
40 39 3expia 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> ( N =/= v -> -. N .<_ ( P .\/ v ) ) )
41 40 necon4ad 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> ( N .<_ ( P .\/ v ) -> N = v ) )
42 30 41 mpd 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N = v )
43 1 2 3 4 5 6 7 8 cdlemg31b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( v e. A /\ F e. T ) ) -> N .<_ ( Q .\/ ( R ` F ) ) )
44 9 10 25 26 27 16 43 syl222anc 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> N .<_ ( Q .\/ ( R ` F ) ) )
45 44 adantr 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> N .<_ ( Q .\/ ( R ` F ) ) )
46 42 45 eqbrtrrd 
 |-  ( ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) /\ N .<_ W ) -> v .<_ ( Q .\/ ( R ` F ) ) )
47 24 46 mtand 
 |-  ( ( ( ( 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 ` P ) =/= P /\ N e. A ) ) -> -. N .<_ W )