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Theorem cdleme4a 30428
Description: Part of proof of Lemma E in [Crawley] p. 114 top.  G represents fs(r). Auxiliary lemma derived from cdleme5 30429. 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 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  HL )
3 hllat 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  Lat )
5 simp21 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  P  e.  A )
6 simp22 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  Q  e.  A )
7 eqid 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29556 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1182 . . 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 980 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  H )
13 simp3 957 . . . . 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 30415 . . . . 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 1189 . . . 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 990 . . . . . 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 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1182 . . . . 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 30187 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 15 . . . . 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 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1182 . . . 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 14156 . . . 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 1182 . . 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 14182 . . 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 1182 . 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 4056 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme18c  30482  cdleme41sn3a  30622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-glb 14109  df-meet 14111  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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