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Theorem dalem20 30428
Description: Lemma for dath 30471. 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 30416 . . . 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 30417 . . . . . 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 2811 . . 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 1593 . . 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 2736 . . 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 2855 . . . 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 2723 . . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   lecple 13529   joincjn 14394   Atomscatm 29999   HLchlt 30086   LPlanesclpl 30227
This theorem is referenced by:  dalem62  30469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-undef 6536  df-riota 6542  df-poset 14396  df-plt 14408  df-lub 14424  df-glb 14425  df-join 14426  df-meet 14427  df-p0 14461  df-lat 14468  df-clat 14530  df-oposet 29912  df-ol 29914  df-oml 29915  df-covers 30002  df-ats 30003  df-atl 30034  df-cvlat 30058  df-hlat 30087  df-llines 30233  df-lplanes 30234
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