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Theorem cdleme1b 29104
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 28242 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
32ad2antrr 709 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  K  e.  Lat )
4 simpr3 968 . . . . 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 28168 . . . . 5  |-  ( R  e.  A  ->  R  e.  B )
84, 7syl 17 . . . 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 29088 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  B )
15143adant3r3 1167 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  U  e.  B )
165, 10latjcl 14000 . . . 4  |-  ( ( K  e.  Lat  /\  R  e.  B  /\  U  e.  B )  ->  ( R  .\/  U
)  e.  B )
173, 8, 15, 16syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( R  .\/  U
)  e.  B )
18 simpr2 967 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  A )
195, 6atbase 28168 . . . . 5  |-  ( Q  e.  A  ->  Q  e.  B )
2018, 19syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  Q  e.  B )
21 simpr1 966 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  A )
225, 6atbase 28168 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
2321, 22syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  P  e.  B )
245, 10latjcl 14000 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  R  e.  B )  ->  ( P  .\/  R
)  e.  B )
253, 23, 8, 24syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( P  .\/  R
)  e.  B )
265, 12lhpbase 28876 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2726ad2antlr 710 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  W  e.  B )
285, 11latmcl 14001 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  B  /\  W  e.  B )  ->  (
( P  .\/  R
)  ./\  W )  e.  B )
293, 25, 27, 28syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( ( P  .\/  R )  ./\  W )  e.  B )
305, 10latjcl 14000 . . . 4  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  ( ( P  .\/  R )  ./\  W )  e.  B )  ->  ( Q  .\/  ( ( P 
.\/  R )  ./\  W ) )  e.  B
)
313, 20, 29, 30syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  -> 
( Q  .\/  (
( P  .\/  R
)  ./\  W )
)  e.  B )
325, 11latmcl 14001 . . 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 1187 . 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 2337 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28142   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme3c  29108  cdleme4a  29117  cdleme5  29118  cdleme7e  29125  cdleme11  29148  cdleme15  29156  cdleme22gb  29172  cdleme19b  29182  cdleme19e  29185  cdleme20d  29190  cdleme20j  29196  cdleme20k  29197  cdleme20l2  29199  cdleme20l  29200  cdleme20m  29201  cdleme22e  29222  cdleme22eALTN  29223  cdleme22f  29224  cdleme27cl  29244  cdlemefr27cl  29281  cdleme35fnpq  29327
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-lat 13996  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lhyp 28866
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