Description: Part of proof of Lemma E in Crawley p. 113. See cdleme3 . (Contributed by NM, 6-Oct-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdleme1.l | |- .<_ = ( le ` K ) |
|
| cdleme1.j | |- .\/ = ( join ` K ) |
||
| cdleme1.m | |- ./\ = ( meet ` K ) |
||
| cdleme1.a | |- A = ( Atoms ` K ) |
||
| cdleme1.h | |- H = ( LHyp ` K ) |
||
| cdleme1.u | |- U = ( ( P .\/ Q ) ./\ W ) |
||
| cdleme1.f | |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
||
| Assertion | cdleme3fa | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> F e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme1.l | |- .<_ = ( le ` K ) |
|
| 2 | cdleme1.j | |- .\/ = ( join ` K ) |
|
| 3 | cdleme1.m | |- ./\ = ( meet ` K ) |
|
| 4 | cdleme1.a | |- A = ( Atoms ` K ) |
|
| 5 | cdleme1.h | |- H = ( LHyp ` K ) |
|
| 6 | cdleme1.u | |- U = ( ( P .\/ Q ) ./\ W ) |
|
| 7 | cdleme1.f | |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
|
| 8 | eqid | |- ( ( P .\/ R ) ./\ W ) = ( ( P .\/ R ) ./\ W ) |
|
| 9 | 1 2 3 4 5 6 7 8 | cdleme3h | |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> F e. A ) |