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Theorem latnlemlt 15377
Description: Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3693 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 15369 . . 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 15372 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
76necon3bbid 2699 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  =/=  X ) )
8 simp1 988 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
91, 3latmcl 15345 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
10 simp2 989 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
11 latnlemlt.s . . . 4  |-  .<  =  ( lt `  K )
122, 11pltval 15253 . . 3  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
)  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
138, 9, 10, 12syl3anc 1219 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   ltcplt 15234   meetcmee 15238   Latclat 15338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-lat 15339
This theorem is referenced by:  hlrelat2  33410
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