Theorem 4atexlemwb

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 ) ) ) )
	4thatlemmwb.h	|- H = ( LHyp ` K )
Assertion	4atexlemwb	|- ( ph -> W e. ( Base ` K ) )

Proof

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		4thatlemmwb.h	|- H = ( LHyp ` K )
3	1	4atexlemw	|- ( ph -> W e. H )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5	4 2	lhpbase	|- ( W e. H -> W e. ( Base ` K ) )
6	3 5	syl	|- ( ph -> W e. ( Base ` K ) )