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Theorem cdlemefs27cl 30895
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of  N. TODO FIX COMMENT This is the start of a re-proof of cdleme27cl 30848 etc. with the  s  .<_  ( P 
.\/  Q ) condition (so as to not have the  C hypothesis). (Contributed by NM, 24-Mar-2013.)
Hypotheses
Ref Expression
cdlemefs26.b  |-  B  =  ( Base `  K
)
cdlemefs26.l  |-  .<_  =  ( le `  K )
cdlemefs26.j  |-  .\/  =  ( join `  K )
cdlemefs26.m  |-  ./\  =  ( meet `  K )
cdlemefs26.a  |-  A  =  ( Atoms `  K )
cdlemefs26.h  |-  H  =  ( LHyp `  K
)
cdlemefs27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefs27.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs27.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemefs27.i  |-  I  =  ( iota_ u  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  u  =  E ) )
cdlemefs27.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
Assertion
Ref Expression
cdlemefs27cl  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  N  e.  B )
Distinct variable groups:    u, t, A    t, B, u    u, E    t, H    t,  .\/ , u   
t, K    t,  .<_ , u   
t,  ./\ , u    t, P, u    t, Q, u    t, U, u    t, W, u   
t, s, u
Allowed substitution hints:    A( s)    B( s)    C( u, t, s)    D( u, t, s)    P( s)    Q( s)    U( s)    E( t, s)    H( u, s)    I( u, t, s)    .\/ ( s)    K( u, s)    .<_ ( s)    ./\ ( s)    N( u, t, s)    W( s)

Proof of Theorem cdlemefs27cl
StepHypRef Expression
1 cdlemefs27.n . 2  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
2 simpr2 964 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  s  .<_  ( P  .\/  Q
) )
3 iftrue 3705 . . . 4  |-  ( s 
.<_  ( P  .\/  Q
)  ->  if (
s  .<_  ( P  .\/  Q ) ,  I ,  C )  =  I )
42, 3syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  C )  =  I )
5 simpl1 960 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simpl2 961 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 simpl3 962 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
8 simpr1 963 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
9 simpr3 965 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  P  =/=  Q )
10 cdlemefs26.b . . . . 5  |-  B  =  ( Base `  K
)
11 cdlemefs26.l . . . . 5  |-  .<_  =  ( le `  K )
12 cdlemefs26.j . . . . 5  |-  .\/  =  ( join `  K )
13 cdlemefs26.m . . . . 5  |-  ./\  =  ( meet `  K )
14 cdlemefs26.a . . . . 5  |-  A  =  ( Atoms `  K )
15 cdlemefs26.h . . . . 5  |-  H  =  ( LHyp `  K
)
16 cdlemefs27.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
17 cdlemefs27.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
18 cdlemefs27.e . . . . 5  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
19 cdlemefs27.i . . . . 5  |-  I  =  ( iota_ u  e.  B A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  u  =  E ) )
2010, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme25cl 30839 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( P  =/= 
Q  /\  s  .<_  ( P  .\/  Q ) ) )  ->  I  e.  B )
215, 6, 7, 8, 9, 2, 20syl312anc 1205 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  I  e.  B )
224, 21eqeltrd 2478 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  C )  e.  B )
231, 22syl5eqel 2488 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  s  .<_  ( P  .\/  Q
)  /\  P  =/=  Q ) )  ->  N  e.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdlemefs29bpre0N  30898  cdlemefs29bpre1N  30899  cdlemefs29cpre1N  30900  cdlemefs29clN  30901  cdlemefs32fvaN  30904  cdlemefs32fva1  30905
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-3or 937  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-rmo 2674  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-iin 4056  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-p1 14424  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  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470
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