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Theorem latnlej2 15561
Description: An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
Hypotheses
Ref Expression
latlej.b  |-  B  =  ( Base `  K
)
latlej.l  |-  .<_  =  ( le `  K )
latlej.j  |-  .\/  =  ( join `  K )
Assertion
Ref Expression
latnlej2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  -.  X  .<_  ( Y  .\/  Z
) )  ->  ( -.  X  .<_  Y  /\  -.  X  .<_  Z ) )

Proof of Theorem latnlej2
StepHypRef Expression
1 latlej.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 latlej.l . . . . . . 7  |-  .<_  =  ( le `  K )
3 latlej.j . . . . . . 7  |-  .\/  =  ( join `  K )
41, 2, 3latlej1 15550 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Y  .<_  ( Y  .\/  Z ) )
543adant3r1 1205 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  .<_  ( Y  .\/  Z
) )
6 simpl 457 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  K  e.  Lat )
7 simpr1 1002 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
8 simpr2 1003 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
91, 3latjcl 15541 . . . . . . 7  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .\/  Z
)  e.  B )
1093adant3r1 1205 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .\/  Z )  e.  B )
111, 2lattr 15546 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  ( Y  .\/  Z
)  e.  B ) )  ->  ( ( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z
) )  ->  X  .<_  ( Y  .\/  Z
) ) )
126, 7, 8, 10, 11syl13anc 1230 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Y  /\  Y  .<_  ( Y  .\/  Z ) )  ->  X  .<_  ( Y  .\/  Z
) ) )
135, 12mpan2d 674 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  X  .<_  ( Y  .\/  Z
) ) )
1413con3d 133 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  ->  -.  X  .<_  Y ) )
151, 2, 3latlej2 15551 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  Z  .<_  ( Y  .\/  Z ) )
16153adant3r1 1205 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  .<_  ( Y  .\/  Z
) )
17 simpr3 1004 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
181, 2lattr 15546 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .\/  Z
)  e.  B ) )  ->  ( ( X  .<_  Z  /\  Z  .<_  ( Y  .\/  Z
) )  ->  X  .<_  ( Y  .\/  Z
) ) )
196, 7, 17, 10, 18syl13anc 1230 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .<_  Z  /\  Z  .<_  ( Y  .\/  Z ) )  ->  X  .<_  ( Y  .\/  Z
) ) )
2016, 19mpan2d 674 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Z  ->  X  .<_  ( Y  .\/  Z
) ) )
2120con3d 133 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  ->  -.  X  .<_  Z ) )
2214, 21jcad 533 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( -.  X  .<_  ( Y 
.\/  Z )  -> 
( -.  X  .<_  Y  /\  -.  X  .<_  Z ) ) )
23223impia 1193 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
)  /\  -.  X  .<_  ( Y  .\/  Z
) )  ->  ( -.  X  .<_  Y  /\  -.  X  .<_  Z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   Latclat 15535
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-lat 15536
This theorem is referenced by:  latnlej2l  15562  latnlej2r  15563
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