Metamath Proof Explorer


Theorem islpln2ah

Description: The predicate "is a lattice plane" for join of atoms. Version of islpln2a expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012)

Ref Expression
Hypotheses islpln2a.l
|- .<_ = ( le ` K )
islpln2a.j
|- .\/ = ( join ` K )
islpln2a.a
|- A = ( Atoms ` K )
islpln2a.p
|- P = ( LPlanes ` K )
islpln2a.y
|- Y = ( ( Q .\/ R ) .\/ S )
Assertion islpln2ah
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )

Proof

Step Hyp Ref Expression
1 islpln2a.l
 |-  .<_ = ( le ` K )
2 islpln2a.j
 |-  .\/ = ( join ` K )
3 islpln2a.a
 |-  A = ( Atoms ` K )
4 islpln2a.p
 |-  P = ( LPlanes ` K )
5 islpln2a.y
 |-  Y = ( ( Q .\/ R ) .\/ S )
6 5 eleq1i
 |-  ( Y e. P <-> ( ( Q .\/ R ) .\/ S ) e. P )
7 1 2 3 4 islpln2a
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )
8 6 7 syl5bb
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )