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 ) )