Metamath Proof Explorer

Theorem dalem28

Description: Lemma for dath . Lemma dalem27 expressed differently. (Contributed by NM, 4-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 ) ) ) )
dalem23.m 
|- ./\ = ( meet ` K )
dalem23.o 
|- O = ( LPlanes ` K )
dalem23.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem23.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem23.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
Assertion dalem28 
|- ( ( ph /\ Y = Z /\ ps ) -> P .<_ ( G .\/ 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 dalem23.m 
 |-  ./\ = ( meet ` K )
7 dalem23.o 
 |-  O = ( LPlanes ` K )
8 dalem23.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem23.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem23.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 1 2 3 4 5 6 7 8 9 10 dalem27 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( G .\/ P ) )
12 1 dalemkehl 
 |-  ( ph -> K e. HL )
13 12 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
14 5 dalemccea 
 |-  ( ps -> c e. A )
15 14 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
16 1 dalempea 
 |-  ( ph -> P e. A )
17 16 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. A )
18 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
19 1 2 3 4 5 6 7 8 9 10 dalem25 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c =/= G )
20 2 3 4 hlatexch1 
 |-  ( ( K e. HL /\ ( c e. A /\ P e. A /\ G e. A ) /\ c =/= G ) -> ( c .<_ ( G .\/ P ) -> P .<_ ( G .\/ c ) ) )
21 13 15 17 18 19 20 syl131anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ ( G .\/ P ) -> P .<_ ( G .\/ c ) ) )
22 11 21 mpd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ ( G .\/ c ) )