Metamath Proof Explorer

Theorem dalem13

Description: Lemma for dalem14 . (Contributed by NM, 21-Jul-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 )
dalem13.o 
|- O = ( LPlanes ` K )
dalem13.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem13.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem13.w 
|- W = ( Y .\/ C )
Assertion dalem13 
|- ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) = W )

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 dalem13.o 
 |-  O = ( LPlanes ` K )
6 dalem13.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 dalem13.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
8 dalem13.w 
 |-  W = ( Y .\/ C )
9 1 dalemkehl 
 |-  ( ph -> K e. HL )
10 9 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> K e. HL )
11 1 dalemyeo 
 |-  ( ph -> Y e. O )
12 11 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> Y e. O )
13 1 dalemzeo 
 |-  ( ph -> Z e. O )
14 13 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> Z e. O )
15 eqid 
 |-  ( LVols ` K ) = ( LVols ` K )
16 1 2 3 4 5 15 6 7 8 dalem9 
 |-  ( ( ph /\ Y =/= Z ) -> W e. ( LVols ` K ) )
17 1 dalemkelat 
 |-  ( ph -> K e. Lat )
18 1 5 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
19 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
20 eqid 
 |-  ( Base ` K ) = ( Base ` K )
21 20 2 3 latlej1 
 |-  ( ( K e. Lat /\ Y e. ( Base ` K ) /\ C e. ( Base ` K ) ) -> Y .<_ ( Y .\/ C ) )
22 17 18 19 21 syl3anc 
 |-  ( ph -> Y .<_ ( Y .\/ C ) )
23 22 8 breqtrrdi 
 |-  ( ph -> Y .<_ W )
24 23 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> Y .<_ W )
25 1 2 3 4 5 6 7 8 dalem8 
 |-  ( ph -> Z .<_ W )
26 25 adantr 
 |-  ( ( ph /\ Y =/= Z ) -> Z .<_ W )
27 simpr 
 |-  ( ( ph /\ Y =/= Z ) -> Y =/= Z )
28 2 3 5 15 2lplnj 
 |-  ( ( K e. HL /\ ( Y e. O /\ Z e. O /\ W e. ( LVols ` K ) ) /\ ( Y .<_ W /\ Z .<_ W /\ Y =/= Z ) ) -> ( Y .\/ Z ) = W )
29 10 12 14 16 24 26 27 28 syl133anc 
 |-  ( ( ph /\ Y =/= Z ) -> ( Y .\/ Z ) = W )