Description: FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat . (Contributed by NM, 27-Mar-2013) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdlemefs29.b | |- B = ( Base ` K ) |
|
| cdlemefs29.l | |- .<_ = ( le ` K ) |
||
| cdlemefs29.j | |- .\/ = ( join ` K ) |
||
| cdlemefs29.m | |- ./\ = ( meet ` K ) |
||
| cdlemefs29.a | |- A = ( Atoms ` K ) |
||
| cdlemefs29.h | |- H = ( LHyp ` K ) |
||
| Assertion | cdlemefs29pre00N | |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ R .<_ ( P .\/ Q ) ) /\ s e. A ) -> ( ( ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemefs29.b | |- B = ( Base ` K ) |
|
| 2 | cdlemefs29.l | |- .<_ = ( le ` K ) |
|
| 3 | cdlemefs29.j | |- .\/ = ( join ` K ) |
|
| 4 | cdlemefs29.m | |- ./\ = ( meet ` K ) |
|
| 5 | cdlemefs29.a | |- A = ( Atoms ` K ) |
|
| 6 | cdlemefs29.h | |- H = ( LHyp ` K ) |
|
| 7 | breq1 | |- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) ) |
|
| 8 | 1 2 3 4 5 6 7 | cdlemefrs29pre00 | |- ( ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ R .<_ ( P .\/ Q ) ) /\ s e. A ) -> ( ( ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) /\ ( s .\/ ( R ./\ W ) ) = R ) <-> ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) ) ) |