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Theorem latnle 14207
Description: Equivalent expressions for "not less than" in a lattice. (chnle 22109 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 14182 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
54biantrurd 494 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =/=  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
61, 2, 3latleeqj1 14185 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
763com23 1157 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
8 eqcom 2298 . . . . 5  |-  ( ( Y  .\/  X )  =  X  <->  X  =  ( Y  .\/  X ) )
97, 8syl6bb 252 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( Y  .\/  X ) ) )
101, 3latjcom 14181 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
1110eqeq2d 2307 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( X  .\/  Y )  <-> 
X  =  ( Y 
.\/  X ) ) )
129, 11bitr4d 247 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( X  .\/  Y ) ) )
1312necon3bbid 2493 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  =/=  ( X 
.\/  Y ) ) )
141, 3latjcl 14172 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 latnle.s . . . 4  |-  .<  =  ( lt `  K )
162, 15pltval 14110 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
1714, 16syld3an3 1227 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
185, 13, 173bitr4d 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   Latclat 14167
This theorem is referenced by:  cvlcvr1  30151  hlrelat  30213  hlrelat2  30214  cvr2N  30222  cvrexchlem  30230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-join 14126  df-lat 14168
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