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Theorem latnlemlt 14513
Description: Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3573 analog.) (Contributed by NM, 5-Feb-2012.)
Hypotheses
Ref Expression
latnlemlt.b  |-  B  =  ( Base `  K
)
latnlemlt.l  |-  .<_  =  ( le `  K )
latnlemlt.s  |-  .<  =  ( lt `  K )
latnlemlt.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latnlemlt  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  .<  X ) )

Proof of Theorem latnlemlt
StepHypRef Expression
1 latnlemlt.b . . . 4  |-  B  =  ( Base `  K
)
2 latnlemlt.l . . . 4  |-  .<_  =  ( le `  K )
3 latnlemlt.m . . . 4  |-  ./\  =  ( meet `  K )
41, 2, 3latmle1 14505 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
54biantrurd 495 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  =/=  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
61, 2, 3latleeqm1 14508 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
76necon3bbid 2635 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  =/=  X ) )
8 simp1 957 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
91, 3latmcl 14480 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
10 simp2 958 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
11 latnlemlt.s . . . 4  |-  .<  =  ( lt `  K )
122, 11pltval 14417 . . 3  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
)  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
138, 9, 10, 12syl3anc 1184 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
145, 7, 133bitr4d 277 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  .<  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   meetcmee 14402   Latclat 14474
This theorem is referenced by:  hlrelat2  30200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-glb 14432  df-meet 14434  df-lat 14475
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