Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  4atlem0a Structured version   Unicode version

Theorem 4atlem0a 30564
Description: Lemma for 4at 30584. (Contributed by NM, 10-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atlem0a  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  -.  R  .<_  ( ( P 
.\/  Q )  .\/  S ) )

Proof of Theorem 4atlem0a
StepHypRef Expression
1 simprr 735 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  -.  S  .<_  ( ( P 
.\/  Q )  .\/  R ) )
2 simpl1 961 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  K  e.  HL )
3 simpl3l 1013 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  R  e.  A )
4 simpl3r 1014 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  S  e.  A )
5 simpl2l 1011 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  P  e.  A )
6 simpl2r 1012 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  Q  e.  A )
7 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . 5  |-  .\/  =  ( join `  K )
9 4at.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 30338 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simprl 734 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
13 4at.l . . . 4  |-  .<_  =  ( le `  K )
147, 13, 8, 9hlexch1 30353 . . 3  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K ) )  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  ( R  .<_  ( ( P  .\/  Q
)  .\/  S )  ->  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
152, 3, 4, 11, 12, 14syl131anc 1198 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( R  .<_  ( ( P 
.\/  Q )  .\/  S )  ->  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )
161, 15mtod 171 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  -.  R  .<_  ( ( P 
.\/  Q )  .\/  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   joincjn 14439   Atomscatm 30235   HLchlt 30322
This theorem is referenced by:  4atlem10  30577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-lat 14513  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323
  Copyright terms: Public domain W3C validator