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Theorem intnatN 32941
Description: If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
intnat.b  |-  B  =  ( Base `  K
)
intnat.l  |-  .<_  =  ( le `  K )
intnat.m  |-  ./\  =  ( meet `  K )
intnat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
intnatN  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( -.  Y  .<_  X  /\  ( X  ./\  Y )  e.  A ) )  ->  -.  Y  e.  A )

Proof of Theorem intnatN
StepHypRef Expression
1 hlatl 32895 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
213ad2ant1 1026 . . . . . 6  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  AtLat )
32ad2antrr 730 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  ( X  ./\  Y )  e.  A )  ->  K  e.  AtLat )
4 eqid 2422 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
5 intnat.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atn0 32843 . . . . 5  |-  ( ( K  e.  AtLat  /\  ( X  ./\  Y )  e.  A )  ->  ( X  ./\  Y )  =/=  ( 0. `  K
) )
73, 6sylancom 671 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  ( X  ./\  Y )  e.  A )  -> 
( X  ./\  Y
)  =/=  ( 0.
`  K ) )
87ex 435 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  ->  ( ( X 
./\  Y )  e.  A  ->  ( X  ./\ 
Y )  =/=  ( 0. `  K ) ) )
9 simpll1 1044 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  K  e.  HL )
10 hllat 32898 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
119, 10syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  K  e.  Lat )
12 simpll2 1045 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  X  e.  B )
13 simpll3 1046 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  Y  e.  B )
14 intnat.b . . . . . . . 8  |-  B  =  ( Base `  K
)
15 intnat.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
1614, 15latmcom 16320 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
1711, 12, 13, 16syl3anc 1264 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
18 simplr 760 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  -.  Y  .<_  X )
199, 1syl 17 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  K  e.  AtLat )
20 simpr 462 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  Y  e.  A )
21 intnat.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
2214, 21, 15, 4, 5atnle 32852 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  Y  e.  A  /\  X  e.  B )  ->  ( -.  Y  .<_  X  <->  ( Y  ./\ 
X )  =  ( 0. `  K ) ) )
2319, 20, 12, 22syl3anc 1264 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  ( -.  Y  .<_  X  <-> 
( Y  ./\  X
)  =  ( 0.
`  K ) ) )
2418, 23mpbid 213 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  ( Y  ./\  X
)  =  ( 0.
`  K ) )
2517, 24eqtrd 2463 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  /\  Y  e.  A )  ->  ( X  ./\  Y
)  =  ( 0.
`  K ) )
2625ex 435 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  ->  ( Y  e.  A  ->  ( X  ./\ 
Y )  =  ( 0. `  K ) ) )
2726necon3ad 2630 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  ->  ( ( X 
./\  Y )  =/=  ( 0. `  K
)  ->  -.  Y  e.  A ) )
288, 27syld 45 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  -.  Y  .<_  X )  ->  ( ( X 
./\  Y )  e.  A  ->  -.  Y  e.  A ) )
2928impr 623 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( -.  Y  .<_  X  /\  ( X  ./\  Y )  e.  A ) )  ->  -.  Y  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   Basecbs 15120   lecple 15196   meetcmee 16189   0.cp0 16282   Latclat 16290   Atomscatm 32798   AtLatcal 32799   HLchlt 32885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-lat 16291  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886
This theorem is referenced by: (None)
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