Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme16aN Structured version   Unicode version

Theorem cdleme16aN 36400
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 1057 . 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 1028 . 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 1029 . 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 1018 . . 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 1019 . . 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 1027 . . 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 1030 . . 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 36185 . . 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 1230 . 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 1031 . . 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 1032 . . 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 2454 . . . 4  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
198, 9, 11, 18lplni2 35677 . . 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 1244 . 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 2454 . . 3  |-  ( ( S  .\/  T ) 
.\/  U )  =  ( ( S  .\/  T )  .\/  U )
229, 11, 18, 21lplnllnneN 35696 . 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 1239 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LPlanesclpl 35632   LHypclh 36124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-p1 15872  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-llines 35638  df-lplanes 35639  df-lhyp 36128
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator