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Theorem dalem18 33683
Description: Lemma for dath 33738. 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 33625 . . 3  |-  ( ph  ->  K  e.  HL )
31dalempea 33628 . . 3  |-  ( ph  ->  P  e.  A )
41dalemqea 33629 . . 3  |-  ( ph  ->  Q  e.  A )
51dalemrea 33630 . . 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 33471 . . 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 1221 . 2  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
11 dalem18.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
1211breq2i 4411 . . . 4  |-  ( c 
.<_  Y  <->  c  .<_  ( ( P  .\/  Q ) 
.\/  R ) )
1312notbii 296 . . 3  |-  ( -.  c  .<_  Y  <->  -.  c  .<_  ( ( P  .\/  Q )  .\/  R ) )
1413rexbii 2862 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Atomscatm 33266   HLchlt 33353
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-p1 15332  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354
This theorem is referenced by:  dalem20  33695
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