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Theorem cdlemefs27cl 29506
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 29459 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 967 . . . 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 3476 . . . 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 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 ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simpl2 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 ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
7 simpl3 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 ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
8 simpr1 966 . . . 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 968 . . . 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 29450 . . . 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 1208 . . 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 2327 . 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 2337 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 939    = wceq 1619    e. wcel 1621    =/= wne 2412   A.wral 2509   ifcif 3470   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdlemefs29bpre0N  29509  cdlemefs29bpre1N  29510  cdlemefs29cpre1N  29511  cdlemefs29clN  29512  cdlemefs32fvaN  29515  cdlemefs32fva1  29516
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081
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