Description: Lemma for 4atexlem7 . (Contributed by NM, 23-Nov-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | 4thatlem.ph | |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
|
| 4thatlempqb.j | |- .\/ = ( join ` K ) |
||
| 4thatlempqb.a | |- A = ( Atoms ` K ) |
||
| Assertion | 4atexlempsb | |- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 4thatlem.ph | |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
|
| 2 | 4thatlempqb.j | |- .\/ = ( join ` K ) |
|
| 3 | 4thatlempqb.a | |- A = ( Atoms ` K ) |
|
| 4 | 1 | 4atexlemk | |- ( ph -> K e. HL ) |
| 5 | 1 | 4atexlemp | |- ( ph -> P e. A ) |
| 6 | 1 | 4atexlems | |- ( ph -> S e. A ) |
| 7 | eqid | |- ( Base ` K ) = ( Base ` K ) |
|
| 8 | 7 2 3 | hlatjcl | |- ( ( K e. HL /\ P e. A /\ S e. A ) -> ( P .\/ S ) e. ( Base ` K ) ) |
| 9 | 4 5 6 8 | syl3anc | |- ( ph -> ( P .\/ S ) e. ( Base ` K ) ) |