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Theorem latnlemlt 15588
Description: Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3735 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 15580 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
54biantrurd 508 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  =/=  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
61, 2, 3latleeqm1 15583 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
76necon3bbid 2714 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  =/=  X ) )
8 simp1 996 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
91, 3latmcl 15556 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
10 simp2 997 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
11 latnlemlt.s . . . 4  |-  .<  =  ( lt `  K )
122, 11pltval 15464 . . 3  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
)  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
138, 9, 10, 12syl3anc 1228 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
145, 7, 133bitr4d 285 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  .<  X ) )
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 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   ltcplt 15445   meetcmee 15449   Latclat 15549
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-lat 15550
This theorem is referenced by:  hlrelat2  34600
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