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Theorem cdleme25cl 30472
Description: Show closure of the unique element in cdleme25c 30470. (Contributed by NM, 2-Feb-2013.)
Hypotheses
Ref Expression
cdleme24.b  |-  B  =  ( Base `  K
)
cdleme24.l  |-  .<_  =  ( le `  K )
cdleme24.j  |-  .\/  =  ( join `  K )
cdleme24.m  |-  ./\  =  ( meet `  K )
cdleme24.a  |-  A  =  ( Atoms `  K )
cdleme24.h  |-  H  =  ( LHyp `  K
)
cdleme24.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme24.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme24.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
cdleme25cl.i  |-  I  =  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
Assertion
Ref Expression
cdleme25cl  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  I  e.  B )
Distinct variable groups:    u, s, A    B, s, u    H, s    .\/ , s, u    K, s   
.<_ , s, u    ./\ , s, u    P, s, u    Q, s, u    R, s, u    W, s, u    u, N    U, s, u
Allowed substitution hints:    F( u, s)    H( u)    I( u, s)    K( u)    N( s)

Proof of Theorem cdleme25cl
StepHypRef Expression
1 cdleme25cl.i . 2  |-  I  =  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
2 cdleme24.b . . . 4  |-  B  =  ( Base `  K
)
3 cdleme24.l . . . 4  |-  .<_  =  ( le `  K )
4 cdleme24.j . . . 4  |-  .\/  =  ( join `  K )
5 cdleme24.m . . . 4  |-  ./\  =  ( meet `  K )
6 cdleme24.a . . . 4  |-  A  =  ( Atoms `  K )
7 cdleme24.h . . . 4  |-  H  =  ( LHyp `  K
)
8 cdleme24.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
9 cdleme24.f . . . 4  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
10 cdleme24.n . . . 4  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  s )  ./\  W
) ) )
112, 3, 4, 5, 6, 7, 8, 9, 10cdleme25c 30470 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  E! u  e.  B  A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N ) )
12 fvex 5683 . . . . 5  |-  ( Base `  K )  e.  _V
132, 12eqeltri 2458 . . . 4  |-  B  e. 
_V
1413riotaclb 6527 . . 3  |-  ( E! u  e.  B  A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N )  <->  ( iota_ u  e.  B A. s  e.  A  ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P 
.\/  Q ) )  ->  u  =  N ) )  e.  B
)
1511, 14sylib 189 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( iota_ u  e.  B A. s  e.  A  (
( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  ->  u  =  N ) )  e.  B )
161, 15syl5eqel 2472 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( P  =/= 
Q  /\  R  .<_  ( P  .\/  Q ) ) )  ->  I  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E!wreu 2652   _Vcvv 2900   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   iota_crio 6479   Basecbs 13397   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   HLchlt 29466   LHypclh 30099
This theorem is referenced by:  cdleme26e  30474  cdleme26eALTN  30476  cdleme26fALTN  30477  cdleme26f  30478  cdleme26f2ALTN  30479  cdleme26f2  30480  cdleme27cl  30481  cdlemefs27cl  30528  cdlemefs32sn1aw  30529  cdleme43fsv1snlem  30535  cdleme41sn3a  30548  cdleme40m  30582  cdleme40n  30583
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-llines 29613  df-lplanes 29614  df-lvols 29615  df-lines 29616  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103
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