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Metamath Proof Explorer

Theorem cdleme20l1

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, penultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t) respectively. (Contributed by NM, 20-Nov-2012)

Ref Expression
Hypotheses cdleme19.l 
|- .<_ = ( le ` K )
cdleme19.j 
|- .\/ = ( join ` K )
cdleme19.m 
|- ./\ = ( meet ` K )
cdleme19.a 
|- A = ( Atoms ` K )
cdleme19.h 
|- H = ( LHyp ` K )
cdleme19.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme20.v 
|- V = ( ( S .\/ T ) ./\ W )
Assertion cdleme20l1 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) e. ( LLines ` K ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l 
 |-  .<_ = ( le ` K )
2 cdleme19.j 
 |-  .\/ = ( join ` K )
3 cdleme19.m 
 |-  ./\ = ( meet ` K )
4 cdleme19.a 
 |-  A = ( Atoms ` K )
5 cdleme19.h 
 |-  H = ( LHyp ` K )
6 cdleme19.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme20.v 
 |-  V = ( ( S .\/ T ) ./\ W )
12 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> K e. HL )
13 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
14 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
16 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> S e. A )
17 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ W )
18 16 17 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
19 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
20 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
21 1 2 3 4 5 6 7 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A )
22 13 14 15 18 19 20 21 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> F e. A )
23 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> W e. H )
24 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R e. A )
25 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> R .<_ ( P .\/ Q ) )
26 1 2 3 4 5 9 cdlemeda 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S e. A /\ -. S .<_ W ) /\ ( R e. A /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D e. A )
27 12 23 16 17 24 25 20 26 syl223anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> D e. A )
28 1 2 3 4 5 6 7 7 9 9 cdleme19c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F =/= D )
29 12 23 14 15 18 24 19 20 28 syl233anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> F =/= D )
30 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
31 2 4 30 llni2 
 |-  ( ( ( K e. HL /\ F e. A /\ D e. A ) /\ F =/= D ) -> ( F .\/ D ) e. ( LLines ` K ) )
32 12 22 27 29 31 syl31anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ S e. A /\ -. S .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) -> ( F .\/ D ) e. ( LLines ` K ) )