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Theorem cdleme4a 33257
Description: Part of proof of Lemma E in [Crawley] p. 114 top.  G represents fs(r). Auxiliary lemma derived from cdleme5 33258. We show fs(r)  <_ p  \/ q. (Contributed by NM, 10-Nov-2012.)
Hypotheses
Ref Expression
cdleme4.l  |-  .<_  =  ( le `  K )
cdleme4.j  |-  .\/  =  ( join `  K )
cdleme4.m  |-  ./\  =  ( meet `  K )
cdleme4.a  |-  A  =  ( Atoms `  K )
cdleme4.h  |-  H  =  ( LHyp `  K
)
cdleme4.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme4.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme4.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme4a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  G  .<_  ( P  .\/  Q
) )

Proof of Theorem cdleme4a
StepHypRef Expression
1 cdleme4.g . 2  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )
2 simp1l 1021 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  HL )
3 hllat 32381 . . . 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 simp21 1030 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  P  e.  A )
6 simp22 1031 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  Q  e.  A )
7 eqid 2402 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 32384 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simp1r 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  H )
13 simp3 999 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  S  e.  A )
14 cdleme4.l . . . . . 6  |-  .<_  =  ( le `  K )
15 cdleme4.m . . . . . 6  |-  ./\  =  ( meet `  K )
16 cdleme4.h . . . . . 6  |-  H  =  ( LHyp `  K
)
17 cdleme4.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme4.f . . . . . 6  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 33244 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 12, 5, 6, 13, 19syl23anc 1237 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  F  e.  ( Base `  K
) )
21 simp23 1032 . . . . . 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 32384 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( R  .\/  S )  e.  ( Base `  K
) )
247, 16lhpbase 33015 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  ( Base `  K
) )
267, 15latmcl 16006 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  (
( R  .\/  S
)  ./\  W )  e.  ( Base `  K
) )
287, 8latjcl 16005 . . . 4  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( R  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
294, 20, 27, 28syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  ( F  .\/  ( ( R 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
307, 14, 15latmle1 16030 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
)  e.  ( Base `  K ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
) )  .<_  ( P 
.\/  Q ) )
314, 11, 29, 30syl3anc 1230 . 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 )
) )  .<_  ( P 
.\/  Q ) )
321, 31syl5eqbr 4428 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  G  .<_  ( P  .\/  Q
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Latclat 15999   Atomscatm 32281   HLchlt 32368   LHypclh 33001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-lat 16000  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-lhyp 33005
This theorem is referenced by:  cdleme18c  33311  cdleme41sn3a  33452
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