Metamath Proof Explorer


Theorem dalemsjteb

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 ) ) ) ) )
dalemb.j
|- .\/ = ( join ` K )
dalemb.a
|- A = ( Atoms ` K )
Assertion dalemsjteb
|- ( ph -> ( S .\/ T ) e. ( Base ` K ) )

Proof

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 dalemb.j
 |-  .\/ = ( join ` K )
3 dalemb.a
 |-  A = ( Atoms ` K )
4 1 dalemkehl
 |-  ( ph -> K e. HL )
5 1 dalemsea
 |-  ( ph -> S e. A )
6 1 dalemtea
 |-  ( ph -> T e. A )
7 eqid
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 3 hlatjcl
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
9 4 5 6 8 syl3anc
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )