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Metamath Proof Explorer

Theorem cdleme0moN

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 9-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` K )
cdleme0.j 
|- .\/ = ( join ` K )
cdleme0.m 
|- ./\ = ( meet ` K )
cdleme0.a 
|- A = ( Atoms ` K )
cdleme0.h 
|- H = ( LHyp ` K )
cdleme0.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme0moN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R = P \/ R = Q ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 cdleme0.h 
 |-  H = ( LHyp ` K )
6 cdleme0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp23r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. R .<_ W )
8 neanior 
 |-  ( ( R =/= P /\ R =/= Q ) <-> -. ( R = P \/ R = Q ) )
9 simpl33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) )
10 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> R e. A )
11 10 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R e. A )
12 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R =/= P )
13 simprr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R =/= Q )
14 simpl32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R .<_ ( P .\/ Q ) )
15 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> K e. HL )
16 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
17 15 16 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> K e. CvLat )
18 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P e. A )
19 18 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> P e. A )
20 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> Q e. A )
21 20 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> Q e. A )
22 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> P =/= Q )
23 4 1 2 cvlsupr2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) )
24 17 19 21 11 22 23 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) )
25 12 13 14 24 mpbir3and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
26 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL )
27 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> W e. H )
28 simp21r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. P .<_ W )
29 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q )
30 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
31 26 27 18 28 20 29 30 syl222anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> U e. A )
32 31 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> U e. A )
33 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( K e. HL /\ W e. H ) )
34 simpl21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( P e. A /\ -. P .<_ W ) )
35 simpl22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
36 1 2 3 4 5 6 cdleme02N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( ( P .\/ U ) = ( Q .\/ U ) /\ U .<_ W ) )
37 36 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P .\/ U ) = ( Q .\/ U ) )
38 33 34 35 22 37 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> ( P .\/ U ) = ( Q .\/ U ) )
39 df-rmo 
 |-  ( E* r e. A ( P .\/ r ) = ( Q .\/ r ) <-> E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) )
40 oveq2 
 |-  ( r = R -> ( P .\/ r ) = ( P .\/ R ) )
41 oveq2 
 |-  ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
42 40 41 eqeq12d 
 |-  ( r = R -> ( ( P .\/ r ) = ( Q .\/ r ) <-> ( P .\/ R ) = ( Q .\/ R ) ) )
43 oveq2 
 |-  ( r = U -> ( P .\/ r ) = ( P .\/ U ) )
44 oveq2 
 |-  ( r = U -> ( Q .\/ r ) = ( Q .\/ U ) )
45 43 44 eqeq12d 
 |-  ( r = U -> ( ( P .\/ r ) = ( Q .\/ r ) <-> ( P .\/ U ) = ( Q .\/ U ) ) )
46 42 45 rmoi 
 |-  ( ( E* r e. A ( P .\/ r ) = ( Q .\/ r ) /\ ( R e. A /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( U e. A /\ ( P .\/ U ) = ( Q .\/ U ) ) ) -> R = U )
47 39 46 syl3an1br 
 |-  ( ( E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) /\ ( R e. A /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( U e. A /\ ( P .\/ U ) = ( Q .\/ U ) ) ) -> R = U )
48 9 11 25 32 38 47 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R = U )
49 36 simprd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> U .<_ W )
50 33 34 35 22 49 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> U .<_ W )
51 48 50 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ ( R =/= P /\ R =/= Q ) ) -> R .<_ W )
52 51 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( R =/= P /\ R =/= Q ) -> R .<_ W ) )
53 8 52 biimtrrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( -. ( R = P \/ R = Q ) -> R .<_ W ) )
54 7 53 mt3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) /\ E* r ( r e. A /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R = P \/ R = Q ) )