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Theorem 4atlem0a 30121
Description: Lemma for 4at 30141. (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 734 . 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 960 . . 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 1012 . . 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 1013 . . 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 1010 . . . 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 1011 . . . 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 2430 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
8 4at.j . . . . 5  |-  .\/  =  ( join `  K )
9 4at.a . . . . 5  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29895 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1184 . . 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 733 . . 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 29910 . . 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 1197 . 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 170 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   joincjn 14384   Atomscatm 29792   HLchlt 29879
This theorem is referenced by:  4atlem10  30134
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-iota 5404  df-fun 5442  df-fv 5448  df-ov 6070  df-lat 14458  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880
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