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Theorem cdleme16aN 29727
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, s  \/ u  =/= t  \/ u. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)
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
cdleme16aN  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  U
)  =/=  ( T 
.\/  U ) )

Proof of Theorem cdleme16aN
StepHypRef Expression
1 simp1ll 1018 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  K  e.  HL )
2 simp22 989 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  S  e.  A )
3 simp23 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  T  e.  A )
4 simp1l 979 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
5 simp1r 980 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
6 simp21 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  Q  e.  A )
7 simp31 991 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  P  =/=  Q )
8 cdleme11.l . . . 4  |-  .<_  =  ( le `  K )
9 cdleme11.j . . . 4  |-  .\/  =  ( join `  K )
10 cdleme11.m . . . 4  |-  ./\  =  ( meet `  K )
11 cdleme11.a . . . 4  |-  A  =  ( Atoms `  K )
12 cdleme11.h . . . 4  |-  H  =  ( LHyp `  K
)
13 cdleme11.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
148, 9, 10, 11, 12, 13lhpat2 29513 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
154, 5, 6, 7, 14syl112anc 1186 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  U  e.  A )
16 simp32 992 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  S  =/=  T )
17 simp33 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  ->  -.  U  .<_  ( S 
.\/  T ) )
18 eqid 2284 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
198, 9, 11, 18lplni2 29005 . . 3  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  ( S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K
) )
201, 2, 3, 15, 16, 17, 19syl132anc 1200 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K
) )
21 eqid 2284 . . 3  |-  ( ( S  .\/  T ) 
.\/  U )  =  ( ( S  .\/  T )  .\/  U )
229, 11, 18, 21lplnllnneN 29024 . 2  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  ( ( S  .\/  T )  .\/  U )  e.  ( LPlanes `  K ) )  -> 
( S  .\/  U
)  =/=  ( T 
.\/  U ) )
231, 2, 3, 15, 20, 22syl131anc 1195 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( P  =/=  Q  /\  S  =/=  T  /\  -.  U  .<_  ( S  .\/  T
) ) )  -> 
( S  .\/  U
)  =/=  ( T 
.\/  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685    =/= wne 2447   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   lecple 13211   joincjn 14074   meetcmee 14075   Atomscatm 28732   HLchlt 28819   LPlanesclpl 28960   LHypclh 29452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-llines 28966  df-lplanes 28967  df-lhyp 29456
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