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Theorem cdlemefr27cl 29743
Description: Part of proof of Lemma E in [Crawley] p. 113. Closure of  N. (Contributed by NM, 23-Mar-2013.)
Hypotheses
Ref Expression
cdlemefr27.b  |-  B  =  ( Base `  K
)
cdlemefr27.l  |-  .<_  =  ( le `  K )
cdlemefr27.j  |-  .\/  =  ( join `  K )
cdlemefr27.m  |-  ./\  =  ( meet `  K )
cdlemefr27.a  |-  A  =  ( Atoms `  K )
cdlemefr27.h  |-  H  =  ( LHyp `  K
)
cdlemefr27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemefr27.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdlemefr27.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
Assertion
Ref Expression
cdlemefr27cl  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  N  e.  B
)

Proof of Theorem cdlemefr27cl
StepHypRef Expression
1 cdlemefr27.n . . 3  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
2 simpr2 967 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  -.  s  .<_  ( P  .\/  Q ) )
3 iffalse 3532 . . . 4  |-  ( -.  s  .<_  ( P  .\/  Q )  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  C )  =  C )
42, 3syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  if ( s 
.<_  ( P  .\/  Q
) ,  I ,  C )  =  C )
51, 4syl5eq 2300 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  N  =  C )
6 simpl1l 1011 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  K  e.  HL )
7 simpl1r 1012 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  W  e.  H
)
8 simpl2 964 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  P  e.  A
)
9 simpl3 965 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  Q  e.  A
)
10 simpr1 966 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  s  e.  A
)
11 cdlemefr27.l . . . 4  |-  .<_  =  ( le `  K )
12 cdlemefr27.j . . . 4  |-  .\/  =  ( join `  K )
13 cdlemefr27.m . . . 4  |-  ./\  =  ( meet `  K )
14 cdlemefr27.a . . . 4  |-  A  =  ( Atoms `  K )
15 cdlemefr27.h . . . 4  |-  H  =  ( LHyp `  K
)
16 cdlemefr27.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
17 cdlemefr27.c . . . 4  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
18 cdlemefr27.b . . . 4  |-  B  =  ( Base `  K
)
1911, 12, 13, 14, 15, 16, 17, 18cdleme1b 29566 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  s  e.  A ) )  ->  C  e.  B )
206, 7, 8, 9, 10, 19syl23anc 1194 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  C  e.  B
)
215, 20eqeltrd 2330 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  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 2419   ifcif 3525   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   meetcmee 14027   Atomscatm 28604   HLchlt 28691   LHypclh 29324
This theorem is referenced by:  cdlemefr29bpre0N  29746  cdlemefr29clN  29747  cdlemefr32fvaN  29749  cdlemefr32fva1  29750
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-lat 14100  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-lhyp 29328
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