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Theorem cdleme1b 30340
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing  F is a lattice element.  F represents their f(r). (Contributed by NM, 6-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l  |-  .<_  =  ( le `  K )
cdleme1.j  |-  .\/  =  ( join `  K )
cdleme1.m  |-  ./\  =  ( meet `  K )
cdleme1.a  |-  A  =  ( Atoms `  K )
cdleme1.h  |-  H  =  ( LHyp `  K
)
cdleme1.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme1.f  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
cdleme1.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme1b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )

Proof of Theorem cdleme1b
StepHypRef Expression
1 cdleme1.f . 2  |-  F  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )
2 hllat 29478 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 707 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  K  e.  Lat )
4 simpr3 965 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  A )
5 cdleme1.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cdleme1.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 29404 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
84, 7syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  R  e.  B )
9 cdleme1.l . . . . . 6  |-  .<_  =  ( le `  K )
10 cdleme1.j . . . . . 6  |-  .\/  =  ( join `  K )
11 cdleme1.m . . . . . 6  |-  ./\  =  ( meet `  K )
12 cdleme1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
13 cdleme1.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
149, 10, 11, 6, 12, 13, 5cdleme0aa 30324 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )
15143adant3r3 1164 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  U  e.  B )
165, 10latjcl 14406 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  B  /\  U  e.  B )  ->  ( R  .\/  U
)  e.  B )
173, 8, 15, 16syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( R  .\/  U
)  e.  B )
18 simpr2 964 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  A )
195, 6atbase 29404 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
2018, 19syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  B )
21 simpr1 963 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  A )
225, 6atbase 29404 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
2321, 22syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  B )
245, 10latjcl 14406 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
253, 23, 8, 24syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( P  .\/  R
)  e.  B )
265, 12lhpbase 30112 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2726ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  W  e.  B )
285, 11latmcl 14407 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  R
)  ./\  W )  e.  B )
293, 25, 27, 28syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( P  .\/  R )  ./\  W )  e.  B )
305, 10latjcl 14406 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  ( ( P  .\/  R )  ./\  W )  e.  B )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)
313, 20, 29, 30syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( Q  .\/  (
( P  .\/  R
)  ./\  W )
)  e.  B )
325, 11latmcl 14407 . . 3  |-  ( ( K  e.  Lat  /\  ( R  .\/  U )  e.  B  /\  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)  ->  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R ) 
./\  W ) ) )  e.  B )
333, 17, 31, 32syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
) )  e.  B
)
341, 33syl5eqel 2471 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   meetcmee 14329   Latclat 14401   Atomscatm 29378   HLchlt 29465   LHypclh 30098
This theorem is referenced by:  cdleme3c  30344  cdleme4a  30353  cdleme5  30354  cdleme7e  30361  cdleme11  30384  cdleme15  30392  cdleme22gb  30408  cdleme19b  30418  cdleme19e  30421  cdleme20d  30426  cdleme20j  30432  cdleme20k  30433  cdleme20l2  30435  cdleme20l  30436  cdleme20m  30437  cdleme22e  30458  cdleme22eALTN  30459  cdleme22f  30460  cdleme27cl  30480  cdlemefr27cl  30517  cdleme35fnpq  30563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-lat 14402  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lhyp 30102
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