Description: A version of hlexchb2 for atoms. (Contributed by NM, 7-Feb-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | hlatexchb.l | |- .<_ = ( le ` K ) |
|
hlatexchb.j | |- .\/ = ( join ` K ) |
||
hlatexchb.a | |- A = ( Atoms ` K ) |
||
Assertion | hlatexchb2 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatexchb.l | |- .<_ = ( le ` K ) |
|
2 | hlatexchb.j | |- .\/ = ( join ` K ) |
|
3 | hlatexchb.a | |- A = ( Atoms ` K ) |
|
4 | hlcvl | |- ( K e. HL -> K e. CvLat ) |
|
5 | 1 2 3 | cvlatexchb2 | |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) ) |
6 | 4 5 | syl3an1 | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) ) |