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Theorem dalem27 34788
Description: Lemma for dath 34825. Show that the line  G P intersects the dummy center of perspectivity  c. (Contributed by NM, 8-Aug-2012.)
Hypotheses
Ref Expression
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
Ref Expression
dalem27  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )

Proof of Theorem dalem27
StepHypRef Expression
1 dalem23.g . . 3  |-  G  =  ( ( c  .\/  P )  ./\  ( d  .\/  S ) )
2 dalem.ph . . . . . 6  |-  ( 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 ) ) ) ) )
32dalemkelat 34713 . . . . 5  |-  ( ph  ->  K  e.  Lat )
433ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
52dalemkehl 34712 . . . . . 6  |-  ( ph  ->  K  e.  HL )
653ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  HL )
7 dalem.ps . . . . . . 7  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
87dalemccea 34772 . . . . . 6  |-  ( ps 
->  c  e.  A
)
983ad2ant3 1019 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
102dalempea 34715 . . . . . 6  |-  ( ph  ->  P  e.  A )
11103ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
12 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
13 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
14 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
1512, 13, 14hlatjcl 34456 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
166, 9, 11, 15syl3anc 1228 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
177dalemddea 34773 . . . . . 6  |-  ( ps 
->  d  e.  A
)
18173ad2ant3 1019 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
192dalemsea 34718 . . . . . 6  |-  ( ph  ->  S  e.  A )
20193ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2112, 13, 14hlatjcl 34456 . . . . 5  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
226, 18, 20, 21syl3anc 1228 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( d  .\/  S
)  e.  ( Base `  K ) )
23 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
24 dalem23.m . . . . 5  |-  ./\  =  ( meet `  K )
2512, 23, 24latmle1 15575 . . . 4  |-  ( ( K  e.  Lat  /\  ( c  .\/  P
)  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  .<_  ( c  .\/  P
) )
264, 16, 22, 25syl3anc 1228 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .<_  ( c  .\/  P
) )
271, 26syl5eqbr 4485 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  .<_  ( c  .\/  P ) )
28 dalem23.o . . . 4  |-  O  =  ( LPlanes `  K )
29 dalem23.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
30 dalem23.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
312, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem23 34785 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
322, 23, 13, 14, 28, 29dalemply 34743 . . . . 5  |-  ( ph  ->  P  .<_  Y )
33323ad2ant1 1017 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 34786 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
35 nbrne2 4470 . . . . 5  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  P  =/=  G
)
3635necomd 2738 . . . 4  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  G  =/=  P
)
3733, 34, 36syl2anc 661 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  P )
3823, 13, 14hlatexch2 34485 . . 3  |-  ( ( K  e.  HL  /\  ( G  e.  A  /\  c  e.  A  /\  P  e.  A
)  /\  G  =/=  P )  ->  ( G  .<_  ( c  .\/  P
)  ->  c  .<_  ( G  .\/  P ) ) )
396, 31, 9, 11, 37, 38syl131anc 1241 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( G  .<_  ( c 
.\/  P )  -> 
c  .<_  ( G  .\/  P ) ) )
4027, 39mpd 15 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  .<_  ( G  .\/  P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   meetcmee 15444   Latclat 15544   Atomscatm 34353   HLchlt 34440   LPlanesclpl 34581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34266  df-ol 34268  df-oml 34269  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-llines 34587  df-lplanes 34588
This theorem is referenced by:  dalem28  34789  dalem32  34793  dalem51  34812  dalem52  34813
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