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Theorem cdleme16aN 30521
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 30307 . . 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 2285 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
198, 9, 11, 18lplni2 29799 . . 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 2285 . . 3  |-  ( ( S  .\/  T ) 
.\/  U )  =  ( ( S  .\/  T )  .\/  U )
229, 11, 18, 21lplnllnneN 29818 . 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 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   lecple 13217   joincjn 14080   meetcmee 14081   Atomscatm 29526   HLchlt 29613   LPlanesclpl 29754   LHypclh 30246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lhyp 30250
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