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Theorem cdleme1b 30708
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 29846 . . . 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 29772 . . . . 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 30692 . . . . 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 14434 . . . 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 29772 . . . . 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 29772 . . . . . . 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 14434 . . . . . 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 30480 . . . . . 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 14435 . . . . 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 14434 . . . 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 14435 . . 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 2488 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 1721   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme3c  30712  cdleme4a  30721  cdleme5  30722  cdleme7e  30729  cdleme11  30752  cdleme15  30760  cdleme22gb  30776  cdleme19b  30786  cdleme19e  30789  cdleme20d  30794  cdleme20j  30800  cdleme20k  30801  cdleme20l2  30803  cdleme20l  30804  cdleme20m  30805  cdleme22e  30826  cdleme22eALTN  30827  cdleme22f  30828  cdleme27cl  30848  cdlemefr27cl  30885  cdleme35fnpq  30931
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-lat 14430  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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