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Theorem dalem20 29882
Description: Lemma for dath 29925. 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 29870 . . . 4  |-  ( ph  ->  E. c  e.  A  -.  c  .<_  Y )
76adantr 451 . . 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 29871 . . . . . 6  |-  ( ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A
)  /\  -.  c  .<_  Y )  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) )
1110ex 423 . . . . 5  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  ->  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) )
1211ancld 536 . . . 4  |-  ( ( ( ph  /\  Y  =  Z )  /\  c  e.  A )  ->  ( -.  c  .<_  Y  -> 
( -.  c  .<_  Y  /\  E. d  e.  A  ( d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
1312reximdva 2655 . . 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 14 . 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 938 . . . . 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 240 . . . 4  |-  ( ps  <->  ( ( c  e.  A  /\  d  e.  A
)  /\  ( -.  c  .<_  Y  /\  (
d  =/=  c  /\  -.  d  .<_  Y  /\  C  .<_  ( c  .\/  d ) ) ) ) )
18172exbii 1570 . . 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 2581 . . 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 2694 . . . 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 2568 . . 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 265 . 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 188 1  |-  ( (
ph  /\  Y  =  Z )  ->  E. c E. d ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  dalem62  29923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
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