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Theorem 3dimlem3 29577
Description: Lemma for 3dim1 29583. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dimlem3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )

Proof of Theorem 3dimlem3
StepHypRef Expression
1 simpr1 963 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  =/=  Q )
2 simpr2 964 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  P  .<_  ( Q  .\/  R ) )
3 simpl11 1032 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  K  e.  HL )
4 simpl2l 1010 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  R  e.  A )
5 simpl12 1033 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  e.  A )
6 simpl13 1034 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  Q  e.  A )
7 simpl3l 1012 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  Q  =/=  R )
87necomd 2635 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  R  =/=  Q )
9 3dim0.l . . . . . 6  |-  .<_  =  ( le `  K )
10 3dim0.j . . . . . 6  |-  .\/  =  ( join `  K )
11 3dim0.a . . . . . 6  |-  A  =  ( Atoms `  K )
129, 10, 11hlatexch2 29512 . . . . 5  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  R  =/=  Q )  ->  ( R  .<_  ( P  .\/  Q
)  ->  P  .<_  ( R  .\/  Q ) ) )
133, 4, 5, 6, 8, 12syl131anc 1197 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( R  .<_  ( P  .\/  Q )  ->  P  .<_  ( R  .\/  Q ) ) )
1410, 11hlatjcom 29484 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
153, 6, 4, 14syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( Q  .\/  R )  =  ( R  .\/  Q
) )
1615breq2d 4167 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( P  .<_  ( Q  .\/  R )  <->  P  .<_  ( R 
.\/  Q ) ) )
1713, 16sylibrd 226 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( R  .<_  ( P  .\/  Q )  ->  P  .<_  ( Q  .\/  R ) ) )
182, 17mtod 170 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
19 simpl1 960 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
20 simpl2 961 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( R  e.  A  /\  S  e.  A )
)
21 simpl3r 1013 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( Q 
.\/  R )  .\/  S ) )
22 simpr3 965 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  P  .<_  ( ( Q  .\/  R )  .\/  S ) )
2310, 9, 113dimlem3a 29576 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( -.  T  .<_  ( ( Q 
.\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R
)  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  R ) )
2419, 20, 21, 2, 22, 23syl113anc 1196 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  R ) )
251, 18, 243jca 1134 1  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( Q  =/=  R  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R
)  .\/  S )
) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   lecple 13465   joincjn 14330   Atomscatm 29380   HLchlt 29467
This theorem is referenced by:  3dim1  29583  3dim2  29584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-join 14362  df-lat 14404  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468
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