Metamath Proof Explorer

Theorem atcvrj1

Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012)

Ref Expression
Hypotheses atcvrj1x.l 
|- .<_ = ( le ` K )
atcvrj1x.j 
|- .\/ = ( join ` K )
atcvrj1x.c 
|- C = (
atcvrj1x.a 
|- A = ( Atoms ` K )
Assertion atcvrj1 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) )

Proof

Step Hyp Ref Expression
1 atcvrj1x.l 
 |-  .<_ = ( le ` K )
2 atcvrj1x.j 
 |-  .\/ = ( join ` K )
3 atcvrj1x.c 
 |-  C = (
4 atcvrj1x.a 
 |-  A = ( Atoms ` K )
5 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P =/= R )
6 hlatl 
 |-  ( K e. HL -> K e. AtLat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> K e. AtLat )
8 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P e. A )
9 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> R e. A )
10 eqid 
 |-  ( meet ` K ) = ( meet ` K )
11 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
12 10 11 4 atnem0 
 |-  ( ( K e. AtLat /\ P e. A /\ R e. A ) -> ( P =/= R <-> ( P ( meet ` K ) R ) = ( 0. ` K ) ) )
13 7 8 9 12 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> ( P =/= R <-> ( P ( meet ` K ) R ) = ( 0. ` K ) ) )
14 5 13 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> ( P ( meet ` K ) R ) = ( 0. ` K ) )
15 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> K e. HL )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
18 8 17 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P e. ( Base ` K ) )
19 16 2 10 11 3 4 cvrp 
 |-  ( ( K e. HL /\ P e. ( Base ` K ) /\ R e. A ) -> ( ( P ( meet ` K ) R ) = ( 0. ` K ) <-> P C ( P .\/ R ) ) )
20 15 18 9 19 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> ( ( P ( meet ` K ) R ) = ( 0. ` K ) <-> P C ( P .\/ R ) ) )
21 14 20 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( P .\/ R ) )
22 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P .<_ ( Q .\/ R ) )
23 1 2 4 hlatexchb2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) )
24 23 3adant3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) )
25 22 24 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
26 21 25 breqtrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) )