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Theorem latnle 15270
Description: Equivalent expressions for "not less than" in a lattice. (chnle 24932 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 15245 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
54biantrurd 508 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =/=  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
61, 2, 3latleeqj1 15248 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
763com23 1193 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
8 eqcom 2445 . . . . 5  |-  ( ( Y  .\/  X )  =  X  <->  X  =  ( Y  .\/  X ) )
97, 8syl6bb 261 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( Y  .\/  X ) ) )
101, 3latjcom 15244 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
1110eqeq2d 2454 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( X  .\/  Y )  <-> 
X  =  ( Y 
.\/  X ) ) )
129, 11bitr4d 256 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( X  .\/  Y ) ) )
1312necon3bbid 2657 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  =/=  ( X 
.\/  Y ) ) )
141, 3latjcl 15236 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 latnle.s . . . 4  |-  .<  =  ( lt `  K )
162, 15pltval 15145 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
1714, 16syld3an3 1263 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
185, 13, 173bitr4d 285 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  .<  ( X  .\/  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4307   ` cfv 5433  (class class class)co 6106   Basecbs 14189   lecple 14260   ltcplt 15126   joincjn 15129   Latclat 15230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-poset 15131  df-plt 15143  df-lub 15159  df-glb 15160  df-join 15161  df-meet 15162  df-lat 15231
This theorem is referenced by:  cvlcvr1  33003  hlrelat  33065  hlrelat2  33066  cvr2N  33074  cvrexchlem  33082
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