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Theorem latnle 16275
Description: Equivalent expressions for "not less than" in a lattice. (chnle 27028 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 16250 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
54biantrurd 510 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =/=  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
61, 2, 3latleeqj1 16253 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
763com23 1211 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  ( Y  .\/  X )  =  X ) )
8 eqcom 2429 . . . . 5  |-  ( ( Y  .\/  X )  =  X  <->  X  =  ( Y  .\/  X ) )
97, 8syl6bb 264 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( Y  .\/  X ) ) )
101, 3latjcom 16249 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  =  ( Y 
.\/  X ) )
1110eqeq2d 2434 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  ( X  .\/  Y )  <-> 
X  =  ( Y 
.\/  X ) ) )
129, 11bitr4d 259 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<_  X  <->  X  =  ( X  .\/  Y ) ) )
1312necon3bbid 2669 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <-> 
X  =/=  ( X 
.\/  Y ) ) )
141, 3latjcl 16241 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
15 latnle.s . . . 4  |-  .<  =  ( lt `  K )
162, 15pltval 16150 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  Y )  e.  B )  -> 
( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
1714, 16syld3an3 1309 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  ( X  .\/  Y )  <->  ( X  .<_  ( X  .\/  Y
)  /\  X  =/=  ( X  .\/  Y ) ) ) )
185, 13, 173bitr4d 288 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   ltcplt 16130   joincjn 16133   Latclat 16235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-lat 16236
This theorem is referenced by:  cvlcvr1  32643  hlrelat  32705  hlrelat2  32706  cvr2N  32714  cvrexchlem  32722
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