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Theorem cdleme17b 33285
Description: Lemma leading to cdleme17c 33286. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l  |-  .<_  =  ( le `  K )
cdleme17.j  |-  .\/  =  ( join `  K )
cdleme17.m  |-  ./\  =  ( meet `  K )
cdleme17.a  |-  A  =  ( Atoms `  K )
cdleme17.h  |-  H  =  ( LHyp `  K
)
cdleme17.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme17.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme17.g  |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme17.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme17b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  C  .<_  ( P  .\/  Q
) )

Proof of Theorem cdleme17b
StepHypRef Expression
1 simp33 1035 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  S  .<_  ( P  .\/  Q
) )
2 eqid 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
3 cdleme17.l . . 3  |-  .<_  =  ( le `  K )
4 simpl1l 1048 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  K  e.  HL )
5 hllat 32361 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  K  e.  Lat )
7 simpl32 1079 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  S  e.  A )
8 cdleme17.a . . . . 5  |-  A  =  ( Atoms `  K )
92, 8atbase 32287 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
107, 9syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  S  e.  ( Base `  K
) )
11 simpl2l 1050 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  P  e.  A )
12 cdleme17.j . . . . 5  |-  .\/  =  ( join `  K )
132, 12, 8hlatjcl 32364 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
144, 11, 7, 13syl3anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
15 simpl31 1078 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  Q  e.  A )
162, 12, 8hlatjcl 32364 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
174, 11, 15, 16syl3anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
183, 12, 8hlatlej2 32373 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
194, 11, 7, 18syl3anc 1230 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  S
) )
20 simpl1r 1049 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  W  e.  H )
21 simpl2r 1051 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  -.  P  .<_  W )
22 cdleme17.m . . . . . 6  |-  ./\  =  ( meet `  K )
23 cdleme17.h . . . . . 6  |-  H  =  ( LHyp `  K
)
24 cdleme17.c . . . . . 6  |-  C  =  ( ( P  .\/  S )  ./\  W )
253, 12, 22, 8, 23, 24cdleme8 33248 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  S  e.  A )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )
264, 20, 11, 21, 7, 25syl221anc 1241 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  ( P  .\/  C )  =  ( P  .\/  S
) )
273, 12, 8hlatlej1 32372 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
284, 11, 15, 27syl3anc 1230 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  P  .<_  ( P  .\/  Q
) )
29 simpr 459 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  C  .<_  ( P  .\/  Q
) )
302, 8atbase 32287 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3111, 30syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  P  e.  ( Base `  K
) )
322, 12, 22, 8, 23, 24cdleme9b 33250 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  ( Base `  K
) )
334, 11, 7, 20, 32syl13anc 1232 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  C  e.  ( Base `  K
) )
342, 3, 12latjle12 16014 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  C  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  Q )  /\  C  .<_  ( P 
.\/  Q ) )  <-> 
( P  .\/  C
)  .<_  ( P  .\/  Q ) ) )
356, 31, 33, 17, 34syl13anc 1232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  (
( P  .<_  ( P 
.\/  Q )  /\  C  .<_  ( P  .\/  Q ) )  <->  ( P  .\/  C )  .<_  ( P 
.\/  Q ) ) )
3628, 29, 35mpbi2and 922 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  ( P  .\/  C )  .<_  ( P  .\/  Q ) )
3726, 36eqbrtrrd 4416 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  ( P  .\/  S )  .<_  ( P  .\/  Q ) )
382, 3, 6, 10, 14, 17, 19, 37lattrd 16010 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  C  .<_  ( P  .\/  Q ) )  ->  S  .<_  ( P  .\/  Q
) )
391, 38mtand 657 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
.\/  Q ) ) )  ->  -.  C  .<_  ( P  .\/  Q
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   meetcmee 15896   Latclat 15997   Atomscatm 32261   HLchlt 32348   LHypclh 32981
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
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 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985
This theorem is referenced by:  cdleme17c  33286
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