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Theorem dalem19 29930
Description: Lemma for dath 29984. Show that a second dummy atom  d exists outside of the  Y and  Z planes (when those planes are equal). (Contributed by NM, 15-Aug-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 )
dalem19.o  |-  O  =  ( LPlanes `  K )
dalem19.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem19.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem19  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
Distinct variable groups:    c, d    A, d    C, d    K, d    .<_ , d    Y, d
Allowed substitution hints:    ph( c, d)    A( c)    C( c)    P( c, d)    Q( c, d)    R( c, d)    S( c, d)    T( c, d)    U( c, d)    .\/ ( c, d)    K( c)    .<_ ( c)    O( c, d)    Y( c)    Z( c, d)

Proof of Theorem dalem19
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 29871 . . 3  |-  ( ph  ->  K  e.  HL )
32ad3antrrr 710 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  K  e.  HL )
4 dalemc.l . . . 4  |-  .<_  =  ( le `  K )
5 dalemc.j . . . 4  |-  .\/  =  ( join `  K )
6 dalemc.a . . . 4  |-  A  =  ( Atoms `  K )
7 dalem19.o . . . 4  |-  O  =  ( LPlanes `  K )
8 dalem19.y . . . 4  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
91, 4, 5, 6, 7, 8dalemcea 29908 . . 3  |-  ( ph  ->  C  e.  A )
109ad3antrrr 710 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  C  e.  A )
11 simplr 731 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  c  e.  A )
121, 7dalemyeb 29897 . . 3  |-  ( ph  ->  Y  e.  ( Base `  K ) )
1312ad3antrrr 710 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  Y  e.  ( Base `  K
) )
14 dalem19.z . . . 4  |-  Z  =  ( ( S  .\/  T )  .\/  U )
151, 4, 5, 6, 7, 8, 14dalem17 29928 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
1615ad2antrr 706 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  C  .<_  Y )
17 simpr 447 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  -.  c  .<_  Y )
18 eqid 2366 . . 3  |-  ( Base `  K )  =  (
Base `  K )
1918, 4, 5, 6atbtwnex 29696 . 2  |-  ( ( ( K  e.  HL  /\  C  e.  A  /\  c  e.  A )  /\  ( Y  e.  (
Base `  K )  /\  C  .<_  Y  /\  -.  c  .<_  Y ) )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
203, 10, 11, 13, 16, 17, 19syl33anc 1198 1  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   Atomscatm 29512   HLchlt 29599   LPlanesclpl 29740
This theorem is referenced by:  dalem20  29941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747
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