Metamath Proof Explorer

Theorem dalem19

Description: Lemma for dath . Show that a second dummy atom d exists outside of the Y and Z planes (when those planes are equal). (Contributed by NM, 15-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 )
dalem19.o 
|- O = ( LPlanes ` K )
dalem19.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem19.z 
|- Z = ( ( S .\/ T ) .\/ U )
Assertion dalem19 
|- ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> E. d e. A ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) )

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 dalem19.o 
 |-  O = ( LPlanes ` K )
6 dalem19.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 dalem19.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
8 1 dalemkehl 
 |-  ( ph -> K e. HL )
9 8 ad3antrrr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> K e. HL )
10 1 2 3 4 5 6 dalemcea 
 |-  ( ph -> C e. A )
11 10 ad3antrrr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> C e. A )
12 simplr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> c e. A )
13 1 5 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
14 13 ad3antrrr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> Y e. ( Base ` K ) )
15 1 2 3 4 5 6 7 dalem17 
 |-  ( ( ph /\ Y = Z ) -> C .<_ Y )
16 15 ad2antrr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> C .<_ Y )
17 simpr 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> -. c .<_ Y )
18 eqid 
 |-  ( Base ` K ) = ( Base ` K )
19 18 2 3 4 atbtwnex 
 |-  ( ( ( K e. HL /\ C e. A /\ c e. A ) /\ ( Y e. ( Base ` K ) /\ C .<_ Y /\ -. c .<_ Y ) ) -> E. d e. A ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) )
20 9 11 12 14 16 17 19 syl33anc 
 |-  ( ( ( ( ph /\ Y = Z ) /\ c e. A ) /\ -. c .<_ Y ) -> E. d e. A ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) )