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Theorem cdleme23a 35020
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 1102 . . . 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 34035 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . 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 1104 . . . . 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 1106 . . . . 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 34038 . . . . 5  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  B )
124, 7, 8, 11syl3anc 1223 . . . 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 1017 . . . . 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 1103 . . . . . 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 34669 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1714, 16syl 16 . . . . 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 15528 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
206, 13, 17, 19syl3anc 1223 . . . 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 15528 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  ./\  ( X  ./\ 
W ) )  e.  B )
226, 12, 20, 21syl3anc 1223 . . 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 15553 . . . 4  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  ./\  ( X  ./\ 
W ) )  .<_  ( X  ./\  W ) )
246, 12, 20, 23syl3anc 1223 . . 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 15553 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
266, 13, 17, 25syl3anc 1223 . . 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 15534 . 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 4473 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521   Atomscatm 33935   HLchlt 34022   LHypclh 34655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-lat 15522  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659
This theorem is referenced by:  cdleme28a  35041
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