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Theorem cdleme16aN 30993
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 1020 . 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 991 . 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 992 . 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 981 . . 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 982 . . 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 990 . . 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 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
) ) )  ->  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 30779 . . 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 1188 . 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 994 . . 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 995 . . 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 2435 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
198, 9, 11, 18lplni2 30271 . . 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 1202 . 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 2435 . . 3  |-  ( ( S  .\/  T ) 
.\/  U )  =  ( ( S  .\/  T )  .\/  U )
229, 11, 18, 21lplnllnneN 30290 . 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 1197 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   lecple 13528   joincjn 14393   meetcmee 14394   Atomscatm 29998   HLchlt 30085   LPlanesclpl 30226   LHypclh 30718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lhyp 30722
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