Metamath Proof Explorer

Theorem cdlemd1

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 29-May-2012)

Ref Expression
Hypotheses cdlemd1.l 
|- .<_ = ( le ` K )
cdlemd1.j 
|- .\/ = ( join ` K )
cdlemd1.m 
|- ./\ = ( meet ` K )
cdlemd1.a 
|- A = ( Atoms ` K )
cdlemd1.h 
|- H = ( LHyp ` K )
Assertion cdlemd1 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R = ( ( P .\/ ( ( P .\/ R ) ./\ W ) ) ./\ ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemd1.l 
 |-  .<_ = ( le ` K )
2 cdlemd1.j 
 |-  .\/ = ( join ` K )
3 cdlemd1.m 
 |-  ./\ = ( meet ` K )
4 cdlemd1.a 
 |-  A = ( Atoms ` K )
5 cdlemd1.h 
 |-  H = ( LHyp ` K )
6 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> K e. HL )
7 simpr1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. A )
8 simpr2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. A )
9 simpr31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. A )
10 simpr32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
11 simpr33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> -. R .<_ ( P .\/ Q ) )
12 1 2 3 4 2llnma2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
13 6 7 8 9 10 11 12 syl132anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
14 hllat 
 |-  ( K e. HL -> K e. Lat )
15 14 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> K e. Lat )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
18 9 17 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R e. ( Base ` K ) )
19 16 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
20 7 19 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> P e. ( Base ` K ) )
21 16 2 latjcom 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( R .\/ P ) = ( P .\/ R ) )
22 15 18 20 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( R .\/ P ) = ( P .\/ R ) )
23 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
24 simpr1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
25 16 1 2 3 4 5 cdlemc1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. ( Base ` K ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( ( P .\/ R ) ./\ W ) ) = ( P .\/ R ) )
26 23 18 24 25 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( P .\/ ( ( P .\/ R ) ./\ W ) ) = ( P .\/ R ) )
27 22 26 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( R .\/ P ) = ( P .\/ ( ( P .\/ R ) ./\ W ) ) )
28 16 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
29 8 28 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> Q e. ( Base ` K ) )
30 16 2 latjcom 
 |-  ( ( K e. Lat /\ R e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( R .\/ Q ) = ( Q .\/ R ) )
31 15 18 29 30 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( R .\/ Q ) = ( Q .\/ R ) )
32 simpr2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
33 16 1 2 3 4 5 cdlemc1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ R e. ( Base ` K ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) = ( Q .\/ R ) )
34 23 18 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) = ( Q .\/ R ) )
35 31 34 eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( R .\/ Q ) = ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) )
36 27 35 oveq12d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = ( ( P .\/ ( ( P .\/ R ) ./\ W ) ) ./\ ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) ) )
37 13 36 eqtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) ) -> R = ( ( P .\/ ( ( P .\/ R ) ./\ W ) ) ./\ ( Q .\/ ( ( Q .\/ R ) ./\ W ) ) ) )