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Theorem latnle 14442
Description: Equivalent expressions for "not less than" in a lattice. (chnle 22865 analog.) (Contributed by NM, 16-Nov-2011.)
Hypotheses
Ref Expression
latnle.b  |-  B  =  ( Base `  K
)
latnle.l  |-  .<_  =  ( le `  K )
latnle.s  |-  .<  =  ( lt `  K )
latnle.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latnle  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  .<  ( X  .\/  Y ) ) )

Proof of Theorem latnle
StepHypRef Expression
1 latnle.b . . . 4  |-  B  =  ( Base `  K
)
2 latnle.l . . . 4  |-  .<_  =  ( le `  K )
3 latnle.j . . . 4  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 14417 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
54biantrurd 495 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =/=  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
61, 2, 3latleeqj1 14420 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
763com23 1159 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
8 eqcom 2390 . . . . 5  |-  ( ( Y  .\/  X )  =  X  <->  X  =  ( Y  .\/  X ) )
97, 8syl6bb 253 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( Y  .\/  X ) ) )
101, 3latjcom 14416 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
1110eqeq2d 2399 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( X  .\/  Y )  <-> 
X  =  ( Y 
.\/  X ) ) )
129, 11bitr4d 248 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( X  .\/  Y ) ) )
1312necon3bbid 2585 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  =/=  ( X 
.\/  Y ) ) )
141, 3latjcl 14407 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 latnle.s . . . 4  |-  .<  =  ( lt `  K )
162, 15pltval 14345 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
1714, 16syld3an3 1229 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
185, 13, 173bitr4d 277 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  .<  ( X  .\/  Y ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Basecbs 13397   lecple 13464   ltcplt 14326   joincjn 14329   Latclat 14402
This theorem is referenced by:  cvlcvr1  29455  hlrelat  29517  hlrelat2  29518  cvr2N  29526  cvrexchlem  29534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-poset 14331  df-plt 14343  df-lub 14359  df-join 14361  df-lat 14403
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