Theorem dalemcnes

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 )
Assertion	dalemcnes	|- ( ph -> C =/= S )

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		dalemc.l	|- .<_ = ( le ` K )
3		dalemc.j	|- .\/ = ( join ` K )
4		dalemc.a	|- A = ( Atoms ` K )
5	1	dalemkelat	|- ( ph -> K e. Lat )
6	1 4	dalemceb	|- ( ph -> C e. ( Base ` K ) )
7	1 4	dalemseb	|- ( ph -> S e. ( Base ` K ) )
8	1 4	dalemteb	|- ( ph -> T e. ( Base ` K ) )
9		simp321	|- ( ( ( ( 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 ) ) ) ) -> -. C .<_ ( S .\/ T ) )
10	1 9	sylbi	|- ( ph -> -. C .<_ ( S .\/ T ) )
11		eqid	|- ( Base ` K ) = ( Base ` K )
12	11 2 3	latnlej1l	|- ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) /\ -. C .<_ ( S .\/ T ) ) -> C =/= S )
13	5 6 7 8 10 12	syl131anc	|- ( ph -> C =/= S )