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Theorem cdlemefr27cl 33964
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 1014 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  -.  s  .<_  ( P  .\/  Q ) )
32iffalsed 3891 . . 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 )
41, 3syl5eq 2496 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  N  =  C )
5 simpl1l 1058 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  K  e.  HL )
6 simpl1r 1059 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  W  e.  H
)
7 simpl2 1011 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  P  e.  A
)
8 simpl3 1012 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  Q  e.  A
)
9 simpr1 1013 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  s  e.  A
)
10 cdlemefr27.l . . . 4  |-  .<_  =  ( le `  K )
11 cdlemefr27.j . . . 4  |-  .\/  =  ( join `  K )
12 cdlemefr27.m . . . 4  |-  ./\  =  ( meet `  K )
13 cdlemefr27.a . . . 4  |-  A  =  ( Atoms `  K )
14 cdlemefr27.h . . . 4  |-  H  =  ( LHyp `  K
)
15 cdlemefr27.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdlemefr27.c . . . 4  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
17 cdlemefr27.b . . . 4  |-  B  =  ( Base `  K
)
1810, 11, 12, 13, 14, 15, 16, 17cdleme1b 33786 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  s  e.  A ) )  ->  C  e.  B )
195, 6, 7, 8, 9, 18syl23anc 1274 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  (
s  e.  A  /\  -.  s  .<_  ( P 
.\/  Q )  /\  P  =/=  Q ) )  ->  C  e.  B
)
204, 19eqeltrd 2528 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   ifcif 3880   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Atomscatm 32823   HLchlt 32910   LHypclh 33543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-lat 16285  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-lhyp 33547
This theorem is referenced by:  cdlemefr29bpre0N  33967  cdlemefr29clN  33968  cdlemefr32fvaN  33970  cdlemefr32fva1  33971
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