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Theorem cdleme17b 33898
Description: Lemma leading to cdleme17c 33899. (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 1052 . 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 2462 . . 3  |-  ( Base `  K )  =  (
Base `  K )
3 cdleme17.l . . 3  |-  .<_  =  ( le `  K )
4 simpl1l 1065 . . . 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 32974 . . . 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 1096 . . . 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 32900 . . . 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 1067 . . . 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 32977 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
144, 11, 7, 13syl3anc 1276 . . 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 1095 . . . 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 32977 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
174, 11, 15, 16syl3anc 1276 . . 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 32986 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
194, 11, 7, 18syl3anc 1276 . . 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 1066 . . . . 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 1068 . . . . 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 33861 . . . . 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 1287 . . . 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 32985 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
284, 11, 15, 27syl3anc 1276 . . . . 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 467 . . . . 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 32900 . . . . . . 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 33863 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  ( Base `  K
) )
334, 11, 7, 20, 32syl13anc 1278 . . . . . 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 16357 . . . . . 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 1278 . . . . 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 937 . . . 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 4439 . . 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 16353 . 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 669 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 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   Basecbs 15170   lecple 15246   joincjn 16238   meetcmee 16239   Latclat 16340   Atomscatm 32874   HLchlt 32961   LHypclh 33594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-preset 16222  df-poset 16240  df-plt 16253  df-lub 16269  df-glb 16270  df-join 16271  df-meet 16272  df-p0 16334  df-p1 16335  df-lat 16341  df-clat 16403  df-oposet 32787  df-ol 32789  df-oml 32790  df-covers 32877  df-ats 32878  df-atl 32909  df-cvlat 32933  df-hlat 32962  df-psubsp 33113  df-pmap 33114  df-padd 33406  df-lhyp 33598
This theorem is referenced by:  cdleme17c  33899
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