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Metamath Proof Explorer

Theorem cdlemd3

Description: Part of proof of Lemma D in Crawley p. 113. The R =/= P requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012)

Ref Expression
Hypotheses cdlemd3.l 
|- .<_ = ( le ` K )
cdlemd3.j 
|- .\/ = ( join ` K )
cdlemd3.a 
|- A = ( Atoms ` K )
cdlemd3.h 
|- H = ( LHyp ` K )
Assertion cdlemd3 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ S ) )

Proof

Step Hyp Ref Expression
1 cdlemd3.l 
 |-  .<_ = ( le ` K )
2 cdlemd3.j 
 |-  .\/ = ( join ` K )
3 cdlemd3.a 
 |-  A = ( Atoms ` K )
4 cdlemd3.h 
 |-  H = ( LHyp ` K )
5 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
7 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
8 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
9 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
10 simp233 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> R =/= P )
11 1 2 3 hlatexch1 
 |-  ( ( K e. HL /\ ( R e. A /\ S e. A /\ P e. A ) /\ R =/= P ) -> ( R .<_ ( P .\/ S ) -> S .<_ ( P .\/ R ) ) )
12 6 7 8 9 10 11 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .<_ ( P .\/ S ) -> S .<_ ( P .\/ R ) ) )
13 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
14 1 2 3 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
15 6 9 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P .<_ ( P .\/ Q ) )
16 simp232 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
17 6 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. Lat )
18 eqid 
 |-  ( Base ` K ) = ( Base ` K )
19 18 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
20 9 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
21 18 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
22 7 21 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. ( Base ` K ) )
23 18 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
24 13 23 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) )
25 18 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
26 17 20 24 25 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
27 18 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ R e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) <-> ( P .\/ R ) .<_ ( P .\/ Q ) ) )
28 17 20 22 26 27 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) <-> ( P .\/ R ) .<_ ( P .\/ Q ) ) )
29 15 16 28 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ R ) .<_ ( P .\/ Q ) )
30 18 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
31 8 30 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. ( Base ` K ) )
32 18 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
33 17 20 22 32 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
34 18 1 lattr 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ ( P .\/ R ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( P .\/ R ) /\ ( P .\/ R ) .<_ ( P .\/ Q ) ) -> S .<_ ( P .\/ Q ) ) )
35 17 31 33 26 34 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( S .<_ ( P .\/ R ) /\ ( P .\/ R ) .<_ ( P .\/ Q ) ) -> S .<_ ( P .\/ Q ) ) )
36 29 35 mpan2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S .<_ ( P .\/ R ) -> S .<_ ( P .\/ Q ) ) )
37 12 36 syld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( R .<_ ( P .\/ S ) -> S .<_ ( P .\/ Q ) ) )
38 5 37 mtod 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ R =/= P ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> -. R .<_ ( P .\/ S ) )