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Theorem dalem61 30592
Description: Lemma for dath 30595. 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 2438 . . 3  |-  ( ( c  .\/  P ) 
./\  ( d  .\/  S ) )  =  ( ( c  .\/  P
)  ./\  ( d  .\/  S ) )
12 eqid 2438 . . 3  |-  ( ( c  .\/  Q ) 
./\  ( d  .\/  T ) )  =  ( ( c  .\/  Q
)  ./\  ( d  .\/  T ) )
13 eqid 2438 . . 3  |-  ( ( c  .\/  R ) 
./\  ( d  .\/  U ) )  =  ( ( c  .\/  R
)  ./\  ( d  .\/  U ) )
14 eqid 2438 . . 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 30590 . 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 30591 . 2  |-  ( (
ph  /\  Y  =  Z  /\  ps )  -> 
( D  .\/  E
)  =  ( ( ( ( ( c 
.\/  P )  ./\  ( d  .\/  S
) )  .\/  (
( c  .\/  Q
)  ./\  ( d  .\/  T ) ) ) 
.\/  ( ( c 
.\/  R )  ./\  ( d  .\/  U
) ) )  ./\  Y ) )
1915, 18breqtrrd 4240 1  |-  ( (
ph  /\  Y  =  Z  /\  ps )  ->  F  .<_  ( D  .\/  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LPlanesclpl 30351
This theorem is referenced by:  dalem62  30593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358  df-lvols 30359
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