Description: TODO: FIX COMMENT. TODO: replace with cdlemg4 . (Contributed by NM, 27-Apr-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdlemg4.l | |- .<_ = ( le ` K ) |
|
| cdlemg4.a | |- A = ( Atoms ` K ) |
||
| cdlemg4.h | |- H = ( LHyp ` K ) |
||
| cdlemg4.t | |- T = ( ( LTrn ` K ) ` W ) |
||
| cdlemg4.r | |- R = ( ( trL ` K ) ` W ) |
||
| cdlemg4.j | |- .\/ = ( join ` K ) |
||
| cdlemg4b.v | |- V = ( R ` G ) |
||
| Assertion | cdlemg6a | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. r .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` r ) ) = r ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdlemg4.l | |- .<_ = ( le ` K ) |
|
| 2 | cdlemg4.a | |- A = ( Atoms ` K ) |
|
| 3 | cdlemg4.h | |- H = ( LHyp ` K ) |
|
| 4 | cdlemg4.t | |- T = ( ( LTrn ` K ) ` W ) |
|
| 5 | cdlemg4.r | |- R = ( ( trL ` K ) ` W ) |
|
| 6 | cdlemg4.j | |- .\/ = ( join ` K ) |
|
| 7 | cdlemg4b.v | |- V = ( R ` G ) |
|
| 8 | 1 2 3 4 5 6 7 | cdlemg4 | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ F e. T ) /\ ( G e. T /\ -. r .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( F ` ( G ` r ) ) = r ) |