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Metamath Proof Explorer

Theorem cdlemb3

Description: Given two atoms not under the fiducial co-atom W , there is a third. Lemma B in Crawley p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle ? Then replace cdlemb2 with it. This is a more general version of cdlemb2 without P =/= Q condition. (Contributed by NM, 27-Apr-2013)

Ref Expression
Hypotheses cdlemg5.l 
|- .<_ = ( le ` K )
cdlemg5.j 
|- .\/ = ( join ` K )
cdlemg5.a 
|- A = ( Atoms ` K )
cdlemg5.h 
|- H = ( LHyp ` K )
Assertion cdlemb3 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg5.l 
 |-  .<_ = ( le ` K )
2 cdlemg5.j 
 |-  .\/ = ( join ` K )
3 cdlemg5.a 
 |-  A = ( Atoms ` K )
4 cdlemg5.h 
 |-  H = ( LHyp ` K )
5 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( K e. HL /\ W e. H ) )
6 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( P e. A /\ -. P .<_ W ) )
7 1 2 3 4 cdlemg5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. r e. A ( P =/= r /\ -. r .<_ W ) )
8 5 6 7 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> E. r e. A ( P =/= r /\ -. r .<_ W ) )
9 ancom 
 |-  ( ( P =/= r /\ -. r .<_ W ) <-> ( -. r .<_ W /\ P =/= r ) )
10 eqcom 
 |-  ( P = r <-> r = P )
11 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> P = Q )
12 11 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( P .\/ P ) = ( P .\/ Q ) )
13 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> K e. HL )
14 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> P e. A )
15 2 3 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
16 13 14 15 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( P .\/ P ) = P )
17 12 16 eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( P .\/ Q ) = P )
18 17 breq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( r .<_ ( P .\/ Q ) <-> r .<_ P ) )
19 hlatl 
 |-  ( K e. HL -> K e. AtLat )
20 13 19 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> K e. AtLat )
21 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> r e. A )
22 1 3 atcmp 
 |-  ( ( K e. AtLat /\ r e. A /\ P e. A ) -> ( r .<_ P <-> r = P ) )
23 20 21 14 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( r .<_ P <-> r = P ) )
24 18 23 bitr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( r = P <-> r .<_ ( P .\/ Q ) ) )
25 10 24 bitrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( P = r <-> r .<_ ( P .\/ Q ) ) )
26 25 necon3abid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( P =/= r <-> -. r .<_ ( P .\/ Q ) ) )
27 26 anbi2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( ( -. r .<_ W /\ P =/= r ) <-> ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) ) )
28 9 27 bitrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q /\ r e. A ) -> ( ( P =/= r /\ -. r .<_ W ) <-> ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) ) )
29 28 3expa 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) /\ r e. A ) -> ( ( P =/= r /\ -. r .<_ W ) <-> ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) ) )
30 29 rexbidva 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> ( E. r e. A ( P =/= r /\ -. r .<_ W ) <-> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) ) )
31 8 30 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P = Q ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )
32 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
33 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
34 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( Q e. A /\ -. Q .<_ W ) )
35 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
36 1 2 3 4 cdlemb2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )
37 32 33 34 35 36 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )
38 31 37 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> E. r e. A ( -. r .<_ W /\ -. r .<_ ( P .\/ Q ) ) )