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Theorem dalem61 35873
Description: Lemma for dath 35876. Show that atoms  D,  E, and  F lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms  c and  d. (Contributed by NM, 11-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
) ) ) )
dalem61.m  |-  ./\  =  ( meet `  K )
dalem61.o  |-  O  =  ( LPlanes `  K )
dalem61.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem61.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
dalem61.d  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
dalem61.e  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
dalem61.f  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
Assertion
Ref Expression
dalem61  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( D  .\/  E ) )

Proof of Theorem dalem61
StepHypRef Expression
1 dalem.ph . . 3  |-  ( 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 . . 3  |-  .<_  =  ( le `  K )
3 dalem.j . . 3  |-  .\/  =  ( join `  K )
4 dalem.a . . 3  |-  A  =  ( Atoms `  K )
5 dalem.ps . . 3  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
6 dalem61.m . . 3  |-  ./\  =  ( meet `  K )
7 dalem61.o . . 3  |-  O  =  ( LPlanes `  K )
8 dalem61.y . . 3  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
9 dalem61.z . . 3  |-  Z  =  ( ( S  .\/  T )  .\/  U )
10 dalem61.f . . 3  |-  F  =  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )
11 eqid 2454 . . 3  |-  ( ( c  .\/  P ) 
./\  ( d  .\/  S ) )  =  ( ( c  .\/  P
)  ./\  ( d  .\/  S ) )
12 eqid 2454 . . 3  |-  ( ( c  .\/  Q ) 
./\  ( d  .\/  T ) )  =  ( ( c  .\/  Q
)  ./\  ( d  .\/  T ) )
13 eqid 2454 . . 3  |-  ( ( c  .\/  R ) 
./\  ( d  .\/  U ) )  =  ( ( c  .\/  R
)  ./\  ( d  .\/  U ) )
14 eqid 2454 . . 3  |-  ( ( ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  .\/  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y )  =  ( ( ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  .\/  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem59 35871 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( ( ( ( ( c  .\/  P )  ./\  ( d  .\/  S ) )  .\/  ( ( c  .\/  Q )  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y ) )
16 dalem61.d . . 3  |-  D  =  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )
17 dalem61.e . . 3  |-  E  =  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )
181, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 11, 12, 13, 14dalem60 35872 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  ( ( ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  .\/  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y ) )
1915, 18breqtrrd 4465 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( D  .\/  E ) )
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   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-3or 972  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  df-lvols 35640
This theorem is referenced by:  dalem62  35874
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