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Theorem dalem18 32662
Description: Lemma for dath 32717. Show that a dummy atom  c exists outside of the  Y and  Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargue's theorem doesn't always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012.)
Hypotheses
Ref Expression
dalema.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 ) ) ) ) )
dalemc.l  |-  .<_  =  ( le `  K )
dalemc.j  |-  .\/  =  ( join `  K )
dalemc.a  |-  A  =  ( Atoms `  K )
dalem18.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
Assertion
Ref Expression
dalem18  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
Distinct variable groups:    A, c    .\/ , c    .<_ , c    P, c    Q, c    R, c
Allowed substitution hints:    ph( c)    C( c)    S( c)    T( c)    U( c)    K( c)    O( c)    Y( c)    Z( c)

Proof of Theorem dalem18
StepHypRef Expression
1 dalema.ph . . . 4  |-  ( 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 ) ) ) ) )
21dalemkehl 32604 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 32607 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 32608 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemrea 32609 . . 3  |-  ( ph  ->  R  e.  A )
6 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
7 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
8 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
96, 7, 83dim3 32450 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
102, 3, 4, 5, 9syl13anc 1230 . 2  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
11 dalem18.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1211breq2i 4400 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1312notbii 294 . . 3  |-  ( -.  c  .<_  Y  <->  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1413rexbii 2903 . 2  |-  ( E. c  e.  A  -.  c  .<_  Y  <->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1510, 14sylibr 212 1  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Atomscatm 32245   HLchlt 32332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-p1 15884  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333
This theorem is referenced by:  dalem20  32674
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