Metamath Proof Explorer

Theorem dalemcjden

Description: Lemma for dath . Show that the dummy atoms form a line. (Contributed by NM, 15-Aug-2012)

Ref Expression
Hypotheses dalem.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 ) ) ) ) )
dalem.l 
|- .<_ = ( le ` K )
dalem.j 
|- .\/ = ( join ` K )
dalem.a 
|- A = ( Atoms ` K )
dalem.ps 
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
Assertion dalemcjden 
|- ( ( ph /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) )

Proof

Step Hyp Ref Expression
1 dalem.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 dalem.l 
 |-  .<_ = ( le ` K )
3 dalem.j 
 |-  .\/ = ( join ` K )
4 dalem.a 
 |-  A = ( Atoms ` K )
5 dalem.ps 
 |-  ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
6 1 dalemkehl 
 |-  ( ph -> K e. HL )
7 6 adantr 
 |-  ( ( ph /\ ps ) -> K e. HL )
8 5 dalemccea 
 |-  ( ps -> c e. A )
9 8 adantl 
 |-  ( ( ph /\ ps ) -> c e. A )
10 5 dalemddea 
 |-  ( ps -> d e. A )
11 10 adantl 
 |-  ( ( ph /\ ps ) -> d e. A )
12 5 dalemccnedd 
 |-  ( ps -> c =/= d )
13 12 adantl 
 |-  ( ( ph /\ ps ) -> c =/= d )
14 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
15 3 4 14 llni2 
 |-  ( ( ( K e. HL /\ c e. A /\ d e. A ) /\ c =/= d ) -> ( c .\/ d ) e. ( LLines ` K ) )
16 7 9 11 13 15 syl31anc 
 |-  ( ( ph /\ ps ) -> ( c .\/ d ) e. ( LLines ` K ) )