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Metamath Proof Explorer

Theorem dalemrotps

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 15-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 ) ) ) )
dalemrotps.y 
|- Y = ( ( P .\/ Q ) .\/ R )
Assertion dalemrotps 
|- ( ( ph /\ ps ) -> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) )

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 dalemrotps.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 5 dalemccea 
 |-  ( ps -> c e. A )
8 5 dalemddea 
 |-  ( ps -> d e. A )
9 7 8 jca 
 |-  ( ps -> ( c e. A /\ d e. A ) )
10 9 adantl 
 |-  ( ( ph /\ ps ) -> ( c e. A /\ d e. A ) )
11 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
12 11 adantl 
 |-  ( ( ph /\ ps ) -> -. c .<_ Y )
13 1 3 4 dalemqrprot 
 |-  ( ph -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
14 6 13 eqtr4id 
 |-  ( ph -> Y = ( ( Q .\/ R ) .\/ P ) )
15 14 breq2d 
 |-  ( ph -> ( c .<_ Y <-> c .<_ ( ( Q .\/ R ) .\/ P ) ) )
16 15 adantr 
 |-  ( ( ph /\ ps ) -> ( c .<_ Y <-> c .<_ ( ( Q .\/ R ) .\/ P ) ) )
17 12 16 mtbid 
 |-  ( ( ph /\ ps ) -> -. c .<_ ( ( Q .\/ R ) .\/ P ) )
18 5 dalemccnedd 
 |-  ( ps -> c =/= d )
19 18 necomd 
 |-  ( ps -> d =/= c )
20 19 adantl 
 |-  ( ( ph /\ ps ) -> d =/= c )
21 5 dalem-ddly 
 |-  ( ps -> -. d .<_ Y )
22 21 adantl 
 |-  ( ( ph /\ ps ) -> -. d .<_ Y )
23 14 breq2d 
 |-  ( ph -> ( d .<_ Y <-> d .<_ ( ( Q .\/ R ) .\/ P ) ) )
24 23 adantr 
 |-  ( ( ph /\ ps ) -> ( d .<_ Y <-> d .<_ ( ( Q .\/ R ) .\/ P ) ) )
25 22 24 mtbid 
 |-  ( ( ph /\ ps ) -> -. d .<_ ( ( Q .\/ R ) .\/ P ) )
26 5 dalemclccjdd 
 |-  ( ps -> C .<_ ( c .\/ d ) )
27 26 adantl 
 |-  ( ( ph /\ ps ) -> C .<_ ( c .\/ d ) )
28 20 25 27 3jca 
 |-  ( ( ph /\ ps ) -> ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) )
29 10 17 28 3jca 
 |-  ( ( ph /\ ps ) -> ( ( c e. A /\ d e. A ) /\ -. c .<_ ( ( Q .\/ R ) .\/ P ) /\ ( d =/= c /\ -. d .<_ ( ( Q .\/ R ) .\/ P ) /\ C .<_ ( c .\/ d ) ) ) )