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Theorem 3dimlem2 34255
Description: Lemma for 3dim1 34263. (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 1024 . 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 1030 . . 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 34164 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
653ad2ant1 1017 . . . . 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 1025 . . . . . 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 1026 . . . . . . 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 1027 . . . . . . 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 1029 . . . . . . 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 1028 . . . . . . 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 34189 . . . . . . 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 1241 . . . . . 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 210 . . . . 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 2508 . . . 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 4459 . . 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 301 . 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 1031 . . 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 6297 . . . 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 4459 . . 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 301 . 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 1176 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   lecple 14558   joincjn 15427   Atomscatm 34060   HLchlt 34147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-lat 15529  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148
This theorem is referenced by:  3dim1  34263  3dim2  34264
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