Metamath Proof Explorer


Theorem atcvrj2

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 atcvrj2
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= 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 1 2 3 4 atcvrj2b
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( Q =/= R /\ P .<_ ( Q .\/ R ) ) <-> P C ( Q .\/ R ) ) )
6 5 biimp3a
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P .<_ ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) )