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Theorem dalem20 30175
Description: Lemma for dath 30218. Show that a second dummy atom  d exists outside of the  Y and  Z planes (when those planes are equal). (Contributed by NM, 14-Aug-2012.)
Hypotheses
Ref Expression
dalem.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 ) ) ) ) )
dalem.l  |-  .<_  =  ( le `  K )
dalem.j  |-  .\/  =  ( join `  K )
dalem.a  |-  A  =  ( Atoms `  K )
dalem.ps  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
dalem20.o  |-  O  =  ( LPlanes `  K )
dalem20.y  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
dalem20.z  |-  Z  =  ( ( S  .\/  T )  .\/  U )
Assertion
Ref Expression
dalem20  |-  ( (
ph  /\  Y  =  Z )  ->  E. c E. d ps )
Distinct variable groups:    c, d, A    C, d    K, d    .<_ , c, d    Y, c, d    .\/ , c    P, c    Q, c    R, c    Z, c    ph, c
Allowed substitution hints:    ph( d)    ps( c, d)    C( c)    P( d)    Q( d)    R( d)    S( c, d)    T( c, d)    U( c, d)    .\/ ( d)    K( c)    O( c, d)    Z( d)

Proof of Theorem dalem20
StepHypRef Expression
1 dalem.ph . . . . 5  |-  ( 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 ) ) ) ) )
2 dalem.l . . . . 5  |-  .<_  =  ( le `  K )
3 dalem.j . . . . 5  |-  .\/  =  ( join `  K )
4 dalem.a . . . . 5  |-  A  =  ( Atoms `  K )
5 dalem20.y . . . . 5  |-  Y  =  ( ( P  .\/  Q )  .\/  R )
61, 2, 3, 4, 5dalem18 30163 . . . 4  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
76adantr 452 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  E. c  e.  A  -.  c  .<_  Y )
8 dalem20.o . . . . . . 7  |-  O  =  ( LPlanes `  K )
9 dalem20.z . . . . . . 7  |-  Z  =  ( ( S  .\/  T )  .\/  U )
101, 2, 3, 4, 8, 5, 9dalem19 30164 . . . . . 6  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
1110ex 424 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
1211ancld 537 . . . 4  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  -> 
( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
1312reximdva 2778 . . 3  |-  ( (
ph  /\  Y  =  Z )  ->  ( E. c  e.  A  -.  c  .<_  Y  ->  E. c  e.  A  ( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
147, 13mpd 15 . 2  |-  ( (
ph  /\  Y  =  Z )  ->  E. c  e.  A  ( -.  c  .<_  Y  /\  E. d  e.  A  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
15 dalem.ps . . . . 5  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) ) )
16 3anass 940 . . . . 5  |-  ( ( ( c  e.  A  /\  d  e.  A
)  /\  -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d
) ) )  <->  ( (
c  e.  A  /\  d  e.  A )  /\  ( -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
1715, 16bitri 241 . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  ( -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
18172exbii 1590 . . 3  |-  ( E. c E. d ps  <->  E. c E. d ( ( c  e.  A  /\  d  e.  A
)  /\  ( -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
19 r2ex 2704 . . 3  |-  ( E. c  e.  A  E. d  e.  A  ( -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )  <->  E. c E. d ( ( c  e.  A  /\  d  e.  A
)  /\  ( -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
20 r19.42v 2822 . . . 4  |-  ( E. d  e.  A  ( -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )  <-> 
( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
2120rexbii 2691 . . 3  |-  ( E. c  e.  A  E. d  e.  A  ( -.  c  .<_  Y  /\  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )  <->  E. c  e.  A  ( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
2218, 19, 213bitr2ri 266 . 2  |-  ( E. c  e.  A  ( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )  <->  E. c E. d ps )
2314, 22sylib 189 1  |-  ( (
ph  /\  Y  =  Z )  ->  E. c E. d ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Atomscatm 29746   HLchlt 29833   LPlanesclpl 29974
This theorem is referenced by:  dalem62  30216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981
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