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Theorem 3dimlem2 30430
Description: Lemma for 3dim1 30438. (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
3dimlem2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S ) ) )

Proof of Theorem 3dimlem2
StepHypRef Expression
1 simp3l 986 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  =/=  Q )
2 simp22 992 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  S  .<_  ( Q  .\/  R ) )
3 3dim0.j . . . . . . 7  |-  .\/  =  ( join `  K )
4 3dim0.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4hlatjcom 30339 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
653ad2ant1 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
7 simp3r 987 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  .<_  ( Q  .\/  R
) )
8 simp11 988 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  K  e.  HL )
9 simp12 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P  e.  A )
10 simp21 991 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  R  e.  A )
11 simp13 990 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  Q  e.  A )
12 3dim0.l . . . . . . . 8  |-  .<_  =  ( le `  K )
1312, 3, 4hlatexchb1 30364 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  ( P  .<_  ( Q  .\/  R
)  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
148, 9, 10, 11, 1, 13syl131anc 1198 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( Q  .\/  P )  =  ( Q 
.\/  R ) ) )
157, 14mpbid 203 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( Q  .\/  P )  =  ( Q  .\/  R
) )
166, 15eqtrd 2475 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  .\/  Q )  =  ( Q  .\/  R
) )
1716breq2d 4255 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( S  .<_  ( P  .\/  Q )  <->  S  .<_  ( Q 
.\/  R ) ) )
182, 17mtbird 294 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
19 simp23 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  T  .<_  ( ( Q 
.\/  R )  .\/  S ) )
2016oveq1d 6132 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
2120breq2d 4255 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( T  .<_  ( ( P 
.\/  Q )  .\/  S )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
2219, 21mtbird 294 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  -.  T  .<_  ( ( P 
.\/  Q )  .\/  S ) )
231, 18, 223jca 1135 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q 
.\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) )  /\  ( P  =/= 
Q  /\  P  .<_  ( Q  .\/  R ) ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   lecple 13574   joincjn 14439   Atomscatm 30235   HLchlt 30322
This theorem is referenced by:  3dim1  30438  3dim2  30439
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-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
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-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  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-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-lub 14469  df-join 14471  df-lat 14513  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323
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