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Theorem latnlemlt 14206
Description: Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3414 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 14198 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
54biantrurd 494 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  =/=  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
61, 2, 3latleeqm1 14201 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
76necon3bbid 2493 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
( X  ./\  Y
)  =/=  X ) )
8 simp1 955 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
91, 3latmcl 14173 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
10 simp2 956 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
11 latnlemlt.s . . . 4  |-  .<  =  ( lt `  K )
122, 11pltval 14110 . . 3  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( X  ./\  Y
)  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
138, 9, 10, 12syl3anc 1182 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  ./\  Y )  .<  X  <->  ( ( X  ./\  Y )  .<_  X  /\  ( X  ./\  Y )  =/=  X ) ) )
145, 7, 133bitr4d 276 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 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   meetcmee 14095   Latclat 14167
This theorem is referenced by:  hlrelat2  30214
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-glb 14125  df-meet 14127  df-lat 14168
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