Metamath Proof Explorer

Theorem dalemrotyz

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 19-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 )
dalemrot.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalemrot.z 
|- Z = ( ( S .\/ T ) .\/ U )
Assertion dalemrotyz 
|- ( ( ph /\ Y = Z ) -> ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) )

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 dalemrot.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
6 dalemrot.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
7 simpr 
 |-  ( ( ph /\ Y = Z ) -> Y = Z )
8 1 3 4 dalemqrprot 
 |-  ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
9 5 8 eqtr4id 
 |-  ( ph -> Y = ( ( Q .\/ R ) .\/ P ) )
10 9 adantr 
 |-  ( ( ph /\ Y = Z ) -> Y = ( ( Q .\/ R ) .\/ P ) )
11 1 dalemkehl 
 |-  ( ph -> K e. HL )
12 1 dalemtea 
 |-  ( ph -> T e. A )
13 1 dalemuea 
 |-  ( ph -> U e. A )
14 1 dalemsea 
 |-  ( ph -> S e. A )
15 3 4 hlatjrot 
 |-  ( ( K e. HL /\ ( T e. A /\ U e. A /\ S e. A ) ) -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) )
16 11 12 13 14 15 syl13anc 
 |-  ( ph -> ( ( T .\/ U ) .\/ S ) = ( ( S .\/ T ) .\/ U ) )
17 6 16 eqtr4id 
 |-  ( ph -> Z = ( ( T .\/ U ) .\/ S ) )
18 17 adantr 
 |-  ( ( ph /\ Y = Z ) -> Z = ( ( T .\/ U ) .\/ S ) )
19 7 10 18 3eqtr3d 
 |-  ( ( ph /\ Y = Z ) -> ( ( Q .\/ R ) .\/ P ) = ( ( T .\/ U ) .\/ S ) )