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Theorem cdleme17b 34294
Description: Lemma leading to cdleme17c 34295. (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 1026 . 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 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
3 cdleme17.l . . 3  |-  .<_  =  ( le `  K )
4 simpl1l 1039 . . . 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 33371 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . 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 1070 . . . 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 33297 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
107, 9syl 16 . . 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 1041 . . . 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 33374 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
144, 11, 7, 13syl3anc 1219 . . 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 1069 . . . 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 33374 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
174, 11, 15, 16syl3anc 1219 . . 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 33383 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
194, 11, 7, 18syl3anc 1219 . . 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 1040 . . . . 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 1042 . . . . 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 34257 . . . . 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 1230 . . . 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 33382 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
284, 11, 15, 27syl3anc 1219 . . . . 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 461 . . . . 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 33297 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
3111, 30syl 16 . . . . . 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 34259 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  W  e.  H
) )  ->  C  e.  ( Base `  K
) )
334, 11, 7, 20, 32syl13anc 1221 . . . . . 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 15355 . . . . . 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 1221 . . . . 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 912 . . . 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 4425 . . 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 15351 . 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 659 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   Atomscatm 33271   HLchlt 33358   LHypclh 33991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-p1 15333  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-psubsp 33510  df-pmap 33511  df-padd 33803  df-lhyp 33995
This theorem is referenced by:  cdleme17c  34295
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