Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalem18 Structured version   Visualization version   Unicode version

Theorem dalem18 33292
Description: Lemma for dath 33347. 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 33234 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 33237 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 33238 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemrea 33239 . . 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 33080 . . 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 1278 . 2  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
11 dalem18.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1211breq2i 4426 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1312notbii 302 . . 3  |-  ( -.  c  .<_  Y  <->  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1413rexbii 2901 . 2  |-  ( E. c  e.  A  -.  c  .<_  Y  <->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1510, 14sylibr 217 1  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   E.wrex 2750   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   Basecbs 15176   lecple 15252   joincjn 16244   Atomscatm 32875   HLchlt 32962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-p1 16341  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963
This theorem is referenced by:  dalem20  33304
  Copyright terms: Public domain W3C validator