Metamath Proof Explorer

Theorem dalem11

Description: Lemma for dath . Analogue of dalem10 for E . (Contributed by NM, 23-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 )
dalem11.m 
|- ./\ = ( meet ` K )
dalem11.o 
|- O = ( LPlanes ` K )
dalem11.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem11.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem11.x 
|- X = ( Y ./\ Z )
dalem11.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
Assertion dalem11 
|- ( ph -> E .<_ X )

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 dalem11.m 
 |-  ./\ = ( meet ` K )
6 dalem11.o 
 |-  O = ( LPlanes ` K )
7 dalem11.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem11.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem11.x 
 |-  X = ( Y ./\ Z )
10 dalem11.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 1 2 3 4 7 8 dalemrot 
 |-  ( ph -> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) )
12 biid 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) )
13 eqid 
 |-  ( ( Q .\/ R ) .\/ P ) = ( ( Q .\/ R ) .\/ P )
14 eqid 
 |-  ( ( T .\/ U ) .\/ S ) = ( ( T .\/ U ) .\/ S )
15 eqid 
 |-  ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) )
16 12 2 3 4 5 6 13 14 15 10 dalem10 
 |-  ( ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( T e. A /\ U e. A /\ S e. A ) ) /\ ( ( ( Q .\/ R ) .\/ P ) e. O /\ ( ( T .\/ U ) .\/ S ) e. O ) /\ ( ( -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) /\ -. C .<_ ( P .\/ Q ) ) /\ ( -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) /\ -. C .<_ ( S .\/ T ) ) /\ ( C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) /\ C .<_ ( P .\/ S ) ) ) ) -> E .<_ ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) )
17 11 16 syl 
 |-  ( ph -> E .<_ ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) )
18 1 3 4 dalemqrprot 
 |-  ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
19 7 18 eqtr4id 
 |-  ( ph -> Y = ( ( Q .\/ R ) .\/ P ) )
20 1 dalemkehl 
 |-  ( ph -> K e. HL )
21 1 dalemtea 
 |-  ( ph -> T e. A )
22 1 dalemuea 
 |-  ( ph -> U e. A )
23 1 dalemsea 
 |-  ( ph -> S e. A )
24 3 4 hlatjrot 
 |-  ( ( K e. HL /\ ( T e. A /\ U e. A /\ S e. A ) ) -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) )
25 20 21 22 23 24 syl13anc 
 |-  ( ph -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) )
26 8 25 eqtr4id 
 |-  ( ph -> Z = ( ( T .\/ U ) .\/ S ) )
27 19 26 oveq12d 
 |-  ( ph -> ( Y ./\ Z ) = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) )
28 9 27 syl5eq 
 |-  ( ph -> X = ( ( ( Q .\/ R ) .\/ P ) ./\ ( ( T .\/ U ) .\/ S ) ) )
29 17 28 breqtrrd 
 |-  ( ph -> E .<_ X )