Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem27 Structured version   Unicode version

Theorem dalem27 35839
Description: Lemma for dath 35876. 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 35764 . . . . 5  |-  ( ph  ->  K  e.  Lat )
433ad2ant1 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  K  e.  Lat )
52dalemkehl 35763 . . . . . 6  |-  ( ph  ->  K  e.  HL )
653ad2ant1 1015 . . . . 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 35823 . . . . . 6  |-  ( ps 
->  c  e.  A
)
983ad2ant3 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
c  e.  A )
102dalempea 35766 . . . . . 6  |-  ( ph  ->  P  e.  A )
11103ad2ant1 1015 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  e.  A )
12 eqid 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
13 dalem.j . . . . . 6  |-  .\/  =  ( join `  K )
14 dalem.a . . . . . 6  |-  A  =  ( Atoms `  K )
1512, 13, 14hlatjcl 35507 . . . . 5  |-  ( ( K  e.  HL  /\  c  e.  A  /\  P  e.  A )  ->  ( c  .\/  P
)  e.  ( Base `  K ) )
166, 9, 11, 15syl3anc 1226 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( c  .\/  P
)  e.  ( Base `  K ) )
177dalemddea 35824 . . . . . 6  |-  ( ps 
->  d  e.  A
)
18173ad2ant3 1017 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
d  e.  A )
192dalemsea 35769 . . . . . 6  |-  ( ph  ->  S  e.  A )
20193ad2ant1 1015 . . . . 5  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  S  e.  A )
2112, 13, 14hlatjcl 35507 . . . . 5  |-  ( ( K  e.  HL  /\  d  e.  A  /\  S  e.  A )  ->  ( d  .\/  S
)  e.  ( Base `  K ) )
226, 18, 20, 21syl3anc 1226 . . . 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 15908 . . . 4  |-  ( ( K  e.  Lat  /\  ( c  .\/  P
)  e.  ( Base `  K )  /\  (
d  .\/  S )  e.  ( Base `  K
) )  ->  (
( c  .\/  P
)  ./\  ( d  .\/  S ) )  .<_  ( c  .\/  P
) )
264, 16, 22, 25syl3anc 1226 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .<_  ( c  .\/  P
) )
271, 26syl5eqbr 4472 . 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 35836 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  e.  A )
322, 23, 13, 14, 28, 29dalemply 35794 . . . . 5  |-  ( ph  ->  P  .<_  Y )
33323ad2ant1 1015 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  P  .<_  Y )
342, 23, 13, 14, 7, 24, 28, 29, 30, 1dalem24 35837 . . . 4  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  -.  G  .<_  Y )
35 nbrne2 4457 . . . . 5  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  P  =/=  G
)
3635necomd 2725 . . . 4  |-  ( ( P  .<_  Y  /\  -.  G  .<_  Y )  ->  G  =/=  P
)
3733, 34, 36syl2anc 659 . . 3  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  G  =/=  P )
3823, 13, 14hlatexch2 35536 . . 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 1239 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   lecple 14794   joincjn 15775   meetcmee 15776   Latclat 15877   Atomscatm 35404   HLchlt 35491   LPlanesclpl 35632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639
This theorem is referenced by:  dalem28  35840  dalem32  35844  dalem51  35863  dalem52  35864
  Copyright terms: Public domain W3C validator