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Theorem latnle 15572
Description: Equivalent expressions for "not less than" in a lattice. (chnle 26136 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 15547 . . 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 15550 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
763com23 1202 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
8 eqcom 2476 . . . . 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 15546 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
1110eqeq2d 2481 . . . 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 2714 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  =/=  ( X 
.\/  Y ) ) )
141, 3latjcl 15538 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 latnle.s . . . 4  |-  .<  =  ( lt `  K )
162, 15pltval 15447 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
1714, 16syld3an3 1273 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   ltcplt 15428   joincjn 15431   Latclat 15532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-lat 15533
This theorem is referenced by:  cvlcvr1  34154  hlrelat  34216  hlrelat2  34217  cvr2N  34225  cvrexchlem  34233
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