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Theorem cdleme11a 30374
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 30384. (Contributed by NM, 12-Jun-2012.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme11a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  .\/  U
)  =  ( S 
.\/  T ) )

Proof of Theorem cdleme11a
StepHypRef Expression
1 simp3rr 1031 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  .<_  ( S  .\/  T ) )
2 simp1l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  K  e.  HL )
3 simp1 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp2l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
5 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( Q  e.  A  /\  P  =/=  Q
) )
6 cdleme11.l . . . . 5  |-  .<_  =  ( le `  K )
7 cdleme11.j . . . . 5  |-  .\/  =  ( join `  K )
8 cdleme11.m . . . . 5  |-  ./\  =  ( meet `  K )
9 cdleme11.a . . . . 5  |-  A  =  ( Atoms `  K )
10 cdleme11.h . . . . 5  |-  H  =  ( LHyp `  K
)
11 cdleme11.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
126, 7, 8, 9, 10, 11lhpat2 30159 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
133, 4, 5, 12syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  e.  A )
14 simp3rl 1030 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  T  e.  A )
15 simp3ll 1028 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  S  e.  A )
16 simp2ll 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  P  e.  A )
17 simp2rl 1026 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  Q  e.  A )
18 simp3l 985 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  e.  A  /\  -.  S  .<_  W ) )
196, 7, 8, 9, 10, 11cdleme0c 30327 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  U  =/=  S )
203, 16, 17, 18, 19syl121anc 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  ->  U  =/=  S )
216, 7, 9hlatexchb1 29507 . . 3  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  T  e.  A  /\  S  e.  A
)  /\  U  =/=  S )  ->  ( U  .<_  ( S  .\/  T
)  <->  ( S  .\/  U )  =  ( S 
.\/  T ) ) )
222, 13, 14, 15, 20, 21syl131anc 1197 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( U  .<_  ( S 
.\/  T )  <->  ( S  .\/  U )  =  ( S  .\/  T ) ) )
231, 22mpbid 202 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q
) )  /\  (
( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  U  .<_  ( S  .\/  T ) ) ) )  -> 
( S  .\/  U
)  =  ( S 
.\/  T ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   meetcmee 14329   Atomscatm 29378   HLchlt 29465   LHypclh 30098
This theorem is referenced by:  cdleme11c  30375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lhyp 30102
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