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Theorem dalem19 30553
Description: Lemma for dath 30607. 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 30494 . . 3  |-  ( ph  ->  K  e.  HL )
32ad3antrrr 712 . 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 30531 . . 3  |-  ( ph  ->  C  e.  A )
109ad3antrrr 712 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  C  e.  A )
11 simplr 733 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  c  e.  A )
121, 7dalemyeb 30520 . . 3  |-  ( ph  ->  Y  e.  ( Base `  K ) )
1312ad3antrrr 712 . 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 30551 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  C  .<_  Y )
1615ad2antrr 708 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  C  .<_  Y )
17 simpr 449 . 2  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  -.  c  .<_  Y )
18 eqid 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
1918, 4, 5, 6atbtwnex 30319 . 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 1200 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363
This theorem is referenced by:  dalem20  30564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370
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