Metamath Proof Explorer


Theorem atcvr1

Description: An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012)

Ref Expression
Hypotheses atcvr1.j
|- .\/ = ( join ` K )
atcvr1.c
|- C = ( 
atcvr1.a
|- A = ( Atoms ` K )
Assertion atcvr1
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Proof

Step Hyp Ref Expression
1 atcvr1.j
 |-  .\/ = ( join ` K )
2 atcvr1.c
 |-  C = ( 
3 atcvr1.a
 |-  A = ( Atoms ` K )
4 hlomcmcv
 |-  ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. CvLat ) )
5 1 2 3 cvlatcvr1
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )
6 4 5 syl3an1
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )