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Theorem dalem20 33177
Description: Lemma for dath 33220. 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 33165 . . . 4  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
76adantr 465 . . 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 33166 . . . . . 6  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
1110ex 434 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
1211ancld 553 . . . 4  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  -> 
( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
1312reximdva 2823 . . 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 969 . . . . 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 249 . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  ( -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
18172exbii 1635 . . 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 2748 . . 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 2870 . . . 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 2735 . . 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 274 . 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 196 1  |-  ( (
ph  /\  Y  =  Z )  ->  E. c E. d ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   Atomscatm 32748   HLchlt 32835   LPlanesclpl 32976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983
This theorem is referenced by:  dalem62  33218
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