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Metamath Proof Explorer

Theorem dalem27

Description: Lemma for dath . Show that the line G P intersects the dummy center of perspectivity c . (Contributed by NM, 8-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 dalem27 
|- ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( G .\/ P ) )

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 dalemkelat 
 |-  ( ph -> K e. Lat )
12 11 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
13 1 dalemkehl 
 |-  ( ph -> K e. HL )
14 13 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
15 5 dalemccea 
 |-  ( ps -> c e. A )
16 15 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
17 1 dalempea 
 |-  ( ph -> P e. A )
18 17 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. A )
19 eqid 
 |-  ( Base ` K ) = ( Base ` K )
20 19 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
21 14 16 18 20 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
22 5 dalemddea 
 |-  ( ps -> d e. A )
23 22 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. A )
24 1 dalemsea 
 |-  ( ph -> S e. A )
25 24 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S e. A )
26 19 3 4 hlatjcl 
 |-  ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
27 14 23 25 26 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
28 19 2 6 latmle1 
 |-  ( ( K e. Lat /\ ( c .\/ P ) e. ( Base ` K ) /\ ( d .\/ S ) e. ( Base ` K ) ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) .<_ ( c .\/ P ) )
29 12 21 27 28 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) .<_ ( c .\/ P ) )
30 10 29 eqbrtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G .<_ ( c .\/ P ) )
31 1 2 3 4 5 6 7 8 9 10 dalem23 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G e. A )
32 1 2 3 4 7 8 dalemply 
 |-  ( ph -> P .<_ Y )
33 32 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P .<_ Y )
34 1 2 3 4 5 6 7 8 9 10 dalem24 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. G .<_ Y )
35 nbrne2 
 |-  ( ( P .<_ Y /\ -. G .<_ Y ) -> P =/= G )
36 35 necomd 
 |-  ( ( P .<_ Y /\ -. G .<_ Y ) -> G =/= P )
37 33 34 36 syl2anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G =/= P )
38 2 3 4 hlatexch2 
 |-  ( ( K e. HL /\ ( G e. A /\ c e. A /\ P e. A ) /\ G =/= P ) -> ( G .<_ ( c .\/ P ) -> c .<_ ( G .\/ P ) ) )
39 14 31 16 18 37 38 syl131anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( G .<_ ( c .\/ P ) -> c .<_ ( G .\/ P ) ) )
40 30 39 mpd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c .<_ ( G .\/ P ) )