Metamath Proof Explorer

Theorem dalemswapyzps

Description: Lemma for dath . Swap the Y and Z planes, along with dummy concurrency (center of perspectivity) atoms c and d , to allow reuse of analogous proofs. (Contributed by NM, 17-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 ) ) ) )
Assertion dalemswapyzps 
|- ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) )

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 5 dalemddea 
 |-  ( ps -> d e. A )
7 5 dalemccea 
 |-  ( ps -> c e. A )
8 6 7 jca 
 |-  ( ps -> ( d e. A /\ c e. A ) )
9 8 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d e. A /\ c e. A ) )
10 5 dalem-ddly 
 |-  ( ps -> -. d .<_ Y )
11 10 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
12 simp2 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y = Z )
13 12 breq2d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .<_ Y <-> d .<_ Z ) )
14 11 13 mtbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Z )
15 5 dalemccnedd 
 |-  ( ps -> c =/= d )
16 15 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c =/= d )
17 5 dalem-ccly 
 |-  ( ps -> -. c .<_ Y )
18 17 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
19 12 breq2d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ Y <-> c .<_ Z ) )
20 18 19 mtbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Z )
21 5 dalemclccjdd 
 |-  ( ps -> C .<_ ( c .\/ d ) )
22 21 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( c .\/ d ) )
23 1 dalemkehl 
 |-  ( ph -> K e. HL )
24 23 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
25 7 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
26 6 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. A )
27 3 4 hlatjcom 
 |-  ( ( K e. HL /\ c e. A /\ d e. A ) -> ( c .\/ d ) = ( d .\/ c ) )
28 24 25 26 27 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) = ( d .\/ c ) )
29 22 28 breqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( d .\/ c ) )
30 16 20 29 3jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) )
31 9 14 30 3jca 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) )