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Theorem cdleme23a 29442
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Dec-2012.)
Hypotheses
Ref Expression
cdleme23.b  |-  B  =  ( Base `  K
)
cdleme23.l  |-  .<_  =  ( le `  K )
cdleme23.j  |-  .\/  =  ( join `  K )
cdleme23.m  |-  ./\  =  ( meet `  K )
cdleme23.a  |-  A  =  ( Atoms `  K )
cdleme23.h  |-  H  =  ( LHyp `  K
)
cdleme23.v  |-  V  =  ( ( S  .\/  T )  ./\  ( X  ./\ 
W ) )
Assertion
Ref Expression
cdleme23a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  .<_  W )

Proof of Theorem cdleme23a
StepHypRef Expression
1 cdleme23.v . 2  |-  V  =  ( ( S  .\/  T )  ./\  ( X  ./\ 
W ) )
2 cdleme23.b . . 3  |-  B  =  ( Base `  K
)
3 cdleme23.l . . 3  |-  .<_  =  ( le `  K )
4 simp11l 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  HL )
5 hllat 28457 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 17 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  Lat )
7 simp12l 1073 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  S  e.  A )
8 simp13l 1075 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  T  e.  A )
9 cdleme23.j . . . . . 6  |-  .\/  =  ( join `  K )
10 cdleme23.a . . . . . 6  |-  A  =  ( Atoms `  K )
112, 9, 10hlatjcl 28460 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  B )
124, 7, 8, 11syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  .\/  T )  e.  B
)
13 simp2l 986 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
14 simp11r 1072 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  H )
15 cdleme23.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
162, 15lhpbase 29091 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1714, 16syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  B )
18 cdleme23.m . . . . . 6  |-  ./\  =  ( meet `  K )
192, 18latmcl 14001 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
206, 13, 17, 19syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
W )  e.  B
)
212, 18latmcl 14001 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  ./\  ( X  ./\ 
W ) )  e.  B )
226, 12, 20, 21syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( X  ./\  W ) )  e.  B )
232, 3, 18latmle2 14027 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  ./\  ( X  ./\ 
W ) )  .<_  ( X  ./\  W ) )
246, 12, 20, 23syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( X  ./\  W ) )  .<_  ( X  ./\ 
W ) )
252, 3, 18latmle2 14027 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
266, 13, 17, 25syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
W )  .<_  W )
272, 3, 6, 22, 20, 17, 24, 26lattrd 14008 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( X  ./\  W ) )  .<_  W )
281, 27syl5eqbr 3953 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  .<_  W )
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   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28357   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdleme28a  29463
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-glb 13953  df-meet 13955  df-lat 13996  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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