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Theorem 2atjlej 35619
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 1032 . . . 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 994 . . . . 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 1027 . . . . 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 1028 . . . . 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 1029 . . . . 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 1030 . . . . 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 1031 . . . . 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 35617 . . . . 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 1244 . . . 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 35567 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q
)  e.  A ) )
152, 3, 4, 14syl3anc 1226 . . . 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 2564 . 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 35567 . . 3  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  =/=  S  <->  -.  ( R  .\/  S
)  e.  A ) )
192, 6, 7, 18syl3anc 1226 . 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 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14794   joincjn 15775   Atomscatm 35404   HLchlt 35491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-clat 15940  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492
This theorem is referenced by:  cdlemg46  36877
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