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Theorem 3dimlem3OLDN 32939
Description: Lemma for 3dim1 32944. (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 1011 . 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 1012 . . 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 1080 . . . . 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 1058 . . . . 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 1081 . . . . 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 1082 . . . . 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 1060 . . . . . 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 2656 . . . . 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 32873 . . . . 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 1277 . . . 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 32845 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
153, 6, 4, 14syl3anc 1264 . . . . 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 4378 . . . 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 237 . . 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 180 . 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 1061 . . 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 32841 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
213, 20syl 17 . . . . . 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 2428 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2322, 11atbase 32767 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
246, 23syl 17 . . . . . 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 32767 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
264, 25syl 17 . . . . . 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 32767 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
285, 27syl 17 . . . . . 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 16289 . . . . . 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 1266 . . . . 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 1013 . . . . . 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 1059 . . . . . . 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 32844 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
343, 6, 4, 33syl3anc 1264 . . . . . . 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 32861 . . . . . . 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 1277 . . . . . 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 213 . . . . 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 2464 . . . 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 4378 . . 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 302 . 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 1185 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   Basecbs 15064   lecple 15140   joincjn 16132   Latclat 16234   Atomscatm 32741   HLchlt 32828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-lat 16235  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829
This theorem is referenced by: (None)
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