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Theorem atlt 34110
Description: Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
atlt.s  |-  .<  =  ( lt `  K )
atlt.j  |-  .\/  =  ( join `  K )
atlt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlt  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P  .\/  Q )  <->  P  =/=  Q ) )

Proof of Theorem atlt
StepHypRef Expression
1 simp1 991 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  HL )
2 simp2 992 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  e.  A )
3 simp3 993 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  e.  A )
4 atlt.s . . . 4  |-  .<  =  ( lt `  K )
5 atlt.j . . . 4  |-  .\/  =  ( join `  K )
6 atlt.a . . . 4  |-  A  =  ( Atoms `  K )
7 eqid 2462 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
84, 5, 6, 7atltcvr 34108 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .<  ( P  .\/  Q )  <->  P (  <o  `  K
) ( P  .\/  Q ) ) )
91, 2, 2, 3, 8syl13anc 1225 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P  .\/  Q )  <->  P (  <o  `  K ) ( P  .\/  Q ) ) )
105, 7, 6atcvr1 34090 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P (  <o  `  K )
( P  .\/  Q
) ) )
119, 10bitr4d 256 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<  ( P  .\/  Q )  <->  P  =/=  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   ltcplt 15419   joincjn 15422    <o ccvr 33936   Atomscatm 33937   HLchlt 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025
This theorem is referenced by:  atlelt  34111  2atlt  34112
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