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Theorem 2atjlej 30277
Description: Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
Hypotheses
Ref Expression
ps1.l  |-  .<_  =  ( le `  K )
ps1.j  |-  .\/  =  ( join `  K )
ps1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atjlej  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  R  =/=  S )

Proof of Theorem 2atjlej
StepHypRef Expression
1 simp33 996 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  ( P  .\/  Q )  .<_  ( R  .\/  S ) )
2 simp1 958 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  K  e.  HL )
3 simp21 991 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  P  e.  A )
4 simp22 992 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  Q  e.  A )
5 simp23 993 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  P  =/=  Q )
6 simp31 994 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  R  e.  A )
7 simp32 995 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  S  e.  A )
8 ps1.l . . . . . 6  |-  .<_  =  ( le `  K )
9 ps1.j . . . . . 6  |-  .\/  =  ( join `  K )
10 ps1.a . . . . . 6  |-  A  =  ( Atoms `  K )
118, 9, 10ps-1 30275 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R  .\/  S ) ) )
122, 3, 4, 5, 6, 7, 11syl132anc 1203 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  (
( P  .\/  Q
)  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R 
.\/  S ) ) )
131, 12mpbid 203 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )
149, 10lnnat 30225 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q
)  e.  A ) )
152, 3, 4, 14syl3anc 1185 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q )  e.  A ) )
165, 15mpbid 203 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  -.  ( P  .\/  Q )  e.  A )
1713, 16eqneltrrd 2531 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  -.  ( R  .\/  S )  e.  A )
189, 10lnnat 30225 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  =/=  S  <->  -.  ( R  .\/  S
)  e.  A ) )
192, 6, 7, 18syl3anc 1185 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  ( R  =/=  S  <->  -.  ( R  .\/  S )  e.  A ) )
2017, 19mpbird 225 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A  /\  ( P 
.\/  Q )  .<_  ( R  .\/  S ) ) )  ->  R  =/=  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   Atomscatm 30062   HLchlt 30149
This theorem is referenced by:  cdlemg46  31533
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150
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