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 ) ) ) ) |
| 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 ) ) ) ) |