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Theorem cdlemefs27cl 29870
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 29823 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 3573 . . . 4  |-  ( s 
.<_  ( P  .\/  Q
)  ->  if (
s  .<_  ( P  .\/  Q ) ,  I ,  C )  =  I )
42, 3syl 17 . . 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 29814 . . . 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 2359 . 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 2369 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 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2448   A.wral 2545   ifcif 3567   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   iota_crio 6291   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   HLchlt 28808   LHypclh 29441
This theorem is referenced by:  cdlemefs29bpre0N  29873  cdlemefs29bpre1N  29874  cdlemefs29cpre1N  29875  cdlemefs29clN  29876  cdlemefs32fvaN  29879  cdlemefs32fva1  29880
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445
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