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Theorem dalem61 33659
Description: Lemma for dath 33662. 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 2450 . . 3  |-  ( ( c  .\/  P ) 
./\  ( d  .\/  S ) )  =  ( ( c  .\/  P
)  ./\  ( d  .\/  S ) )
12 eqid 2450 . . 3  |-  ( ( c  .\/  Q ) 
./\  ( d  .\/  T ) )  =  ( ( c  .\/  Q
)  ./\  ( d  .\/  T ) )
13 eqid 2450 . . 3  |-  ( ( c  .\/  R ) 
./\  ( d  .\/  U ) )  =  ( ( c  .\/  R
)  ./\  ( d  .\/  U ) )
14 eqid 2450 . . 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 33657 . 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 33658 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  ( ( ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  .\/  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y ) )
1915, 18breqtrrd 4402 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Atomscatm 33190   HLchlt 33277   LPlanesclpl 33418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-lvols 33426
This theorem is referenced by:  dalem62  33660
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