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Theorem cdleme16aN 29698
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 1023 . 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 994 . 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 995 . 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 984 . . 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 985 . . 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 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
) ) )  ->  Q  e.  A )
7 simp31 996 . . 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 29484 . . 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 1191 . 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 997 . . 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 998 . . 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 2258 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
198, 9, 11, 18lplni2 28976 . . 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 1205 . 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 2258 . . 3  |-  ( ( S  .\/  T ) 
.\/  U )  =  ( ( S  .\/  T )  .\/  U )
229, 11, 18, 21lplnllnneN 28995 . 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 1200 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 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   lecple 13178   joincjn 14041   meetcmee 14042   Atomscatm 28703   HLchlt 28790   LPlanesclpl 28931   LHypclh 29423
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lhyp 29427
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