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Metamath Proof Explorer

Theorem dalem48

Description: Lemma for dath . Analogue of dalem45 for P Q . (Contributed by NM, 16-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 ) ) ) )
dalem44.m 
|- ./\ = ( meet ` K )
dalem44.o 
|- O = ( LPlanes ` K )
dalem44.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem44.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem44.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
dalem44.h 
|- H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
dalem44.i 
|- I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
Assertion dalem48 
|- ( ( ph /\ ps ) -> -. c .<_ ( P .\/ Q ) )

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 dalem44.m 
 |-  ./\ = ( meet ` K )
7 dalem44.o 
 |-  O = ( LPlanes ` K )
8 dalem44.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem44.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem44.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 dalem44.h 
 |-  H = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
12 dalem44.i 
 |-  I = ( ( c .\/ R ) ./\ ( d .\/ U ) )
13 1 dalemkelat 
 |-  ( ph -> K e. Lat )
14 13 adantr 
 |-  ( ( ph /\ ps ) -> K e. Lat )
15 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
16 15 adantl 
 |-  ( ( ph /\ ps ) -> c e. ( Base ` K ) )
17 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
18 17 adantr 
 |-  ( ( ph /\ ps ) -> ( P .\/ Q ) e. ( Base ` K ) )
19 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
20 19 adantr 
 |-  ( ( ph /\ ps ) -> R e. ( Base ` K ) )
21 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
22 8 breq2i 
 |-  ( c .<_ Y <-> c .<_ ( ( P .\/ Q ) .\/ R ) )
23 21 22 sylnib 
 |-  ( ps -> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
24 23 adantl 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
25 eqid 
 |-  ( Base ` K ) = ( Base ` K )
26 25 2 3 latnlej2l 
 |-  ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) /\ -. c .<_ ( ( P .\/ Q ) .\/ R ) ) -> -. c .<_ ( P .\/ Q ) )
27 14 16 18 20 24 26 syl131anc 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( P .\/ Q ) )