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Theorem cdlemeda 29391
Description: Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 13-Nov-2012.)
Hypotheses
Ref Expression
cdlemeda.l  |-  .<_  =  ( le `  K )
cdlemeda.j  |-  .\/  =  ( join `  K )
cdlemeda.m  |-  ./\  =  ( meet `  K )
cdlemeda.a  |-  A  =  ( Atoms `  K )
cdlemeda.h  |-  H  =  ( LHyp `  K
)
cdlemeda.d  |-  D  =  ( ( R  .\/  S )  ./\  W )
Assertion
Ref Expression
cdlemeda  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  D  e.  A )

Proof of Theorem cdlemeda
StepHypRef Expression
1 cdlemeda.d . 2  |-  D  =  ( ( R  .\/  S )  ./\  W )
2 simp1l 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
3 simp31 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  e.  A )
4 simp2l 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  e.  A )
5 cdlemeda.j . . . . . 6  |-  .\/  =  ( join `  K )
6 cdlemeda.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6hlatjcom 28461 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  =  ( S 
.\/  R ) )
82, 3, 4, 7syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( R  .\/  S
)  =  ( S 
.\/  R ) )
98oveq1d 5725 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  =  ( ( S 
.\/  R )  ./\  W ) )
10 simp1r 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  W  e.  H )
11 simp2r 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  S  .<_  W )
12 simp32 997 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  R  .<_  ( P  .\/  Q ) )
13 simp33 998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
14 cdlemeda.l . . . . . 6  |-  .<_  =  ( le `  K )
15 cdlemeda.h . . . . . 6  |-  H  =  ( LHyp `  K
)
1614, 5, 6, 15cdlemesner 29389 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  =/=  R )
172, 3, 4, 12, 13, 16syl122anc 1196 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  S  =/=  R )
18 cdlemeda.m . . . . 5  |-  ./\  =  ( meet `  K )
1914, 5, 18, 6, 15lhpat 29136 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  S  =/=  R ) )  ->  ( ( S 
.\/  R )  ./\  W )  e.  A )
202, 10, 4, 11, 3, 17, 19syl222anc 1203 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( S  .\/  R )  ./\  W )  e.  A )
219, 20eqeltrd 2327 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( R  .\/  S )  ./\  W )  e.  A )
221, 21syl5eqel 2337 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  D  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdlemednpq  29392  cdleme19d  29399  cdleme20aN  29402  cdleme20c  29404  cdleme20f  29407  cdleme20g  29408  cdleme20j  29411  cdleme20l1  29413  cdleme20l2  29414
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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