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Theorem cdleme4a 30725
Description: Part of proof of Lemma E in [Crawley] p. 114 top.  G represents fs(r). Auxiliary lemma derived from cdleme5 30726. 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 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  HL )
3 hllat 29850 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  K  e.  Lat )
5 simp21 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  P  e.  A )
6 simp22 991 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  Q  e.  A )
7 eqid 2408 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme4.j . . . . 5  |-  .\/  =  ( join `  K )
9 cdleme4.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29853 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1184 . . 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 982 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  S  e.  A )  ->  W  e.  H )
13 simp3 959 . . . . 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 30712 . . . . 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 1191 . . . 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 992 . . . . . 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 29853 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1184 . . . . 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 30484 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 16 . . . . 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 14439 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( R  .\/  S )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1184 . . . 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 14438 . . . 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 1184 . . 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 14464 . . 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 1184 . 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 4209 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Latclat 14433   Atomscatm 29750   HLchlt 29837   LHypclh 30470
This theorem is referenced by:  cdleme18c  30779  cdleme41sn3a  30919
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-glb 14391  df-meet 14393  df-lat 14434  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-lhyp 30474
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