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Theorem dalem18 30492
Description: Lemma for dath 30547. 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 30434 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 30437 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 30438 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemrea 30439 . . 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 30280 . . 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 1184 . 2  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
11 dalem18.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1211breq2i 4047 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1312notbii 287 . . 3  |-  ( -.  c  .<_  Y  <->  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1413rexbii 2581 . 2  |-  ( E. c  e.  A  -.  c  .<_  Y  <->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1510, 14sylibr 203 1  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  dalem20  30504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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