Metamath Proof Explorer

Theorem atcvr0eq

Description: The covers relation is not transitive. ( atcv0eq analog.) (Contributed by NM, 29-Nov-2011)

Ref Expression
Hypotheses atcvr0eq.j 
|- .\/ = ( join ` K )
atcvr0eq.z 
|- .0. = ( 0. ` K )
atcvr0eq.c 
|- C = (
atcvr0eq.a 
|- A = ( Atoms ` K )
Assertion atcvr0eq 
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) <-> P = Q ) )

Proof

Step Hyp Ref Expression
1 atcvr0eq.j 
 |-  .\/ = ( join ` K )
2 atcvr0eq.z 
 |-  .0. = ( 0. ` K )
3 atcvr0eq.c 
 |-  C = (
4 atcvr0eq.a 
 |-  A = ( Atoms ` K )
5 1 3 4 atcvr1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )
6 2 3 4 atcvr0 
 |-  ( ( K e. HL /\ P e. A ) -> .0. C P )
7 6 3adant3 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. C P )
8 7 biantrurd 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P C ( P .\/ Q ) <-> ( .0. C P /\ P C ( P .\/ Q ) ) ) )
9 5 8 bitrd 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( .0. C P /\ P C ( P .\/ Q ) ) ) )
10 simp1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. HL )
11 hlop 
 |-  ( K e. HL -> K e. OP )
12 11 3ad2ant1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. OP )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 op0cl 
 |-  ( K e. OP -> .0. e. ( Base ` K ) )
15 12 14 syl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. e. ( Base ` K ) )
16 13 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
17 16 3ad2ant2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. ( Base ` K ) )
18 13 1 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
19 13 3 cvrntr 
 |-  ( ( K e. HL /\ ( .0. e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( .0. C P /\ P C ( P .\/ Q ) ) -> -. .0. C ( P .\/ Q ) ) )
20 10 15 17 18 19 syl13anc 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( .0. C P /\ P C ( P .\/ Q ) ) -> -. .0. C ( P .\/ Q ) ) )
21 9 20 sylbid 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q -> -. .0. C ( P .\/ Q ) ) )
22 21 necon4ad 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) -> P = Q ) )
23 1 4 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
24 23 3adant3 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ P ) = P )
25 7 24 breqtrrd 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> .0. C ( P .\/ P ) )
26 oveq2 
 |-  ( P = Q -> ( P .\/ P ) = ( P .\/ Q ) )
27 26 breq2d 
 |-  ( P = Q -> ( .0. C ( P .\/ P ) <-> .0. C ( P .\/ Q ) ) )
28 25 27 syl5ibcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P = Q -> .0. C ( P .\/ Q ) ) )
29 22 28 impbid 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( .0. C ( P .\/ Q ) <-> P = Q ) )