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Theorem dalem27 35125
Description: Lemma for dath 35162. 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 35050 . . . . 5  |-  ( ph  ->  K  e.  Lat )
433ad2ant1 1016 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
52dalemkehl 35049 . . . . . 6  |-  ( ph  ->  K  e.  HL )
653ad2ant1 1016 . . . . 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 35109 . . . . . 6  |-  ( ps 
->  c  e.  A
)
983ad2ant3 1018 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
102dalempea 35052 . . . . . 6  |-  ( ph  ->  P  e.  A )
11103ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
12 eqid 2441 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
13 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
14 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
1512, 13, 14hlatjcl 34793 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
166, 9, 11, 15syl3anc 1227 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
177dalemddea 35110 . . . . . 6  |-  ( ps 
->  d  e.  A
)
18173ad2ant3 1018 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
192dalemsea 35055 . . . . . 6  |-  ( ph  ->  S  e.  A )
20193ad2ant1 1016 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2112, 13, 14hlatjcl 34793 . . . . 5  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
226, 18, 20, 21syl3anc 1227 . . . 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 1227 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .<_  ( c  .\/  P
) )
271, 26syl5eqbr 4466 . 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 35122 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
322, 23, 13, 14, 28, 29dalemply 35080 . . . . 5  |-  ( ph  ->  P  .<_  Y )
33323ad2ant1 1016 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 35123 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
35 nbrne2 4451 . . . . 5  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  P  =/=  G
)
3635necomd 2712 . . . 4  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  G  =/=  P
)
3733, 34, 36syl2anc 661 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  P )
3823, 13, 14hlatexch2 34822 . . 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 1240 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   lecple 14576   joincjn 15442   meetcmee 15443   Latclat 15544   Atomscatm 34690   HLchlt 34777   LPlanesclpl 34918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-preset 15426  df-poset 15444  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 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925
This theorem is referenced by:  dalem28  35126  dalem32  35130  dalem51  35149  dalem52  35150
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