Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemefr27cl Unicode version

Theorem cdlemefr27cl 29860
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 964 . . . 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 3574 . . . 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 2329 . 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 1008 . . 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 1009 . . 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 961 . . 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 962 . . 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 963 . . 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 29683 . . 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 1191 . 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 2359 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 936    = wceq 1624    e. wcel 1685    =/= wne 2448   ifcif 3567   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Atomscatm 28721   HLchlt 28808   LHypclh 29441
This theorem is referenced by:  cdlemefr29bpre0N  29863  cdlemefr29clN  29864  cdlemefr32fvaN  29866  cdlemefr32fva1  29867
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-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  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-fv 5230  df-ov 5823  df-lat 14147  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-lhyp 29445
  Copyright terms: Public domain W3C validator