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Theorem cdleme22gb 29613
Description: Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.)
Hypotheses
Ref Expression
cdleme18d.l  |-  .<_  =  ( le `  K )
cdleme18d.j  |-  .\/  =  ( join `  K )
cdleme18d.m  |-  ./\  =  ( meet `  K )
cdleme18d.a  |-  A  =  ( Atoms `  K )
cdleme18d.h  |-  H  =  ( LHyp `  K
)
cdleme18d.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme18d.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme18d.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
cdleme22.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
cdleme22gb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  G  e.  B )

Proof of Theorem cdleme22gb
StepHypRef Expression
1 cdleme18d.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  K  e.  HL )
3 hllat 28683 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  K  e.  Lat )
5 simp2l 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  P  e.  A )
6 simp2r 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  Q  e.  A )
7 cdleme22.b . . . . 5  |-  B  =  ( Base `  K
)
8 cdleme18d.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme18d.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 28686 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
112, 5, 6, 10syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( P  .\/  Q )  e.  B
)
12 simp1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp3r 989 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  S  e.  A )
14 cdleme18d.l . . . . . 6  |-  .<_  =  ( le `  K )
15 cdleme18d.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 cdleme18d.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdleme18d.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme18d.f . . . . . 6  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 29545 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  B )
2012, 5, 6, 13, 19syl13anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  F  e.  B )
21 simp3l 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  R  e.  A )
227, 8, 9hlatjcl 28686 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  B )
232, 21, 13, 22syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( R  .\/  S )  e.  B
)
24 simp1r 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  W  e.  H )
257, 16lhpbase 29317 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
2624, 25syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  W  e.  B )
277, 15latmcl 14084 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  B  /\  W  e.  B )  ->  (
( R  .\/  S
)  ./\  W )  e.  B )
284, 23, 26, 27syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  B )
297, 8latjcl 14083 . . . 4  |-  ( ( K  e.  Lat  /\  F  e.  B  /\  ( ( R  .\/  S )  ./\  W )  e.  B )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  B
)
304, 20, 28, 29syl3anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( F  .\/  ( ( R  .\/  S )  ./\  W )
)  e.  B )
317, 15latmcl 14084 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  B
)  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  B )
324, 11, 30, 31syl3anc 1187 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S ) 
./\  W ) ) )  e.  B )
331, 32syl5eqel 2340 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )
)  ->  G  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   meetcmee 14006   Latclat 14078   Atomscatm 28583   HLchlt 28670   LHypclh 29303
This theorem is referenced by:  cdleme25a  29672  cdleme25dN  29675
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-lat 14079  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671  df-lhyp 29307
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