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Theorem 3dimlem3OLDN 35329
Description: Lemma for 3dim1 35334. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dimlem3OLDN  |-  ( ( ( ( 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 3dimlem3OLDN
StepHypRef Expression
1 simpr1 1002 . 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 1003 . . 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 1071 . . . . 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 1049 . . . . 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 1072 . . . . 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 1073 . . . . 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 1051 . . . . . 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 2728 . . . . 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 35263 . . . . 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 1241 . . . 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 35235 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
153, 6, 4, 14syl3anc 1228 . . . . 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 4468 . . . 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 234 . . 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 177 . 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 simpl3r 1052 . . 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 ) )
20 hllat 35231 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
213, 20syl 16 . . . . . 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 )
) )  ->  K  e.  Lat )
22 eqid 2457 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2322, 11atbase 35157 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
246, 23syl 16 . . . . . 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  e.  ( Base `  K
) )
2522, 11atbase 35157 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
264, 25syl 16 . . . . . 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 )
) )  ->  R  e.  ( Base `  K
) )
2722, 11atbase 35157 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
285, 27syl 16 . . . . . 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 )
) )  ->  P  e.  ( Base `  K
) )
2922, 10latjrot 15857 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) ) )  -> 
( ( Q  .\/  R )  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
3021, 24, 26, 28, 29syl13anc 1230 . . . . 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
)  .\/  P )  =  ( ( P 
.\/  Q )  .\/  R ) )
31 simpr3 1004 . . . . . 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 )
) )  ->  P  .<_  ( ( Q  .\/  R )  .\/  S ) )
32 simpl2r 1050 . . . . . . 7  |-  ( ( ( ( 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 )
) )  ->  S  e.  A )
3322, 10, 11hlatjcl 35234 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
343, 6, 4, 33syl3anc 1228 . . . . . . 7  |-  ( ( ( ( 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 )  e.  ( Base `  K
) )
3522, 9, 10, 11hlexchb1 35251 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  ( Q  .\/  R
)  e.  ( Base `  K ) )  /\  -.  P  .<_  ( Q 
.\/  R ) )  ->  ( P  .<_  ( ( Q  .\/  R
)  .\/  S )  <->  ( ( Q  .\/  R
)  .\/  P )  =  ( ( Q 
.\/  R )  .\/  S ) ) )
363, 5, 32, 34, 2, 35syl131anc 1241 . . . . . 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 )
) )  ->  ( P  .<_  ( ( Q 
.\/  R )  .\/  S )  <->  ( ( Q 
.\/  R )  .\/  P )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
3731, 36mpbid 210 . . . . 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
)  .\/  P )  =  ( ( Q 
.\/  R )  .\/  S ) )
3830, 37eqtr3d 2500 . . . 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 )  =  ( ( Q 
.\/  R )  .\/  S ) )
3938breq2d 4468 . . 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  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  T  .<_  ( ( Q  .\/  R ) 
.\/  S ) ) )
4019, 39mtbird 301 . 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 ) )
411, 18, 403jca 1176 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   Latclat 15802   Atomscatm 35131   HLchlt 35218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-covers 35134  df-ats 35135  df-atl 35166  df-cvlat 35190  df-hlat 35219
This theorem is referenced by: (None)
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