Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | dalema.ph | |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
|
| dalemc.l | |- .<_ = ( le ` K ) |
||
| dalemc.j | |- .\/ = ( join ` K ) |
||
| dalemc.a | |- A = ( Atoms ` K ) |
||
| dalempnes.o | |- O = ( LPlanes ` K ) |
||
| dalempnes.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
||
| Assertion | dalempjsen | |- ( ph -> ( P .\/ S ) e. ( LLines ` K ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dalema.ph | |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) ) |
|
| 2 | dalemc.l | |- .<_ = ( le ` K ) |
|
| 3 | dalemc.j | |- .\/ = ( join ` K ) |
|
| 4 | dalemc.a | |- A = ( Atoms ` K ) |
|
| 5 | dalempnes.o | |- O = ( LPlanes ` K ) |
|
| 6 | dalempnes.y | |- Y = ( ( P .\/ Q ) .\/ R ) |
|
| 7 | 1 | dalemkehl | |- ( ph -> K e. HL ) |
| 8 | 1 | dalempea | |- ( ph -> P e. A ) |
| 9 | 1 | dalemsea | |- ( ph -> S e. A ) |
| 10 | 1 2 3 4 5 6 | dalempnes | |- ( ph -> P =/= S ) |
| 11 | eqid | |- ( LLines ` K ) = ( LLines ` K ) |
|
| 12 | 3 4 11 | llni2 | |- ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ P =/= S ) -> ( P .\/ S ) e. ( LLines ` K ) ) |
| 13 | 7 8 9 10 12 | syl31anc | |- ( ph -> ( P .\/ S ) e. ( LLines ` K ) ) |