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Theorem 2atjlej 33429
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 1026 . . . 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 988 . . . . 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 1021 . . . . 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 1022 . . . . 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 1023 . . . . 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 1024 . . . . 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 1025 . . . . 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 33427 . . . . 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 1237 . . . 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 210 . . 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 33377 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q
)  e.  A ) )
152, 3, 4, 14syl3anc 1219 . . . 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 210 . . 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 2561 . 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 33377 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  =/=  S  <->  -.  ( R  .\/  S
)  e.  A ) )
192, 6, 7, 18syl3anc 1219 . 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 232 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   lecple 14347   joincjn 15216   Atomscatm 33214   HLchlt 33301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302
This theorem is referenced by:  cdlemg46  34685
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