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Theorem 2llnmeqat 32552
Description: An atom equals the intersection of two majorizing lines. (Contributed by NM, 3-Apr-2013.)
Hypotheses
Ref Expression
2llnmeqat.l  |-  .<_  =  ( le `  K )
2llnmeqat.m  |-  ./\  =  ( meet `  K )
2llnmeqat.a  |-  A  =  ( Atoms `  K )
2llnmeqat.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
2llnmeqat  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  =  ( X  ./\  Y ) )

Proof of Theorem 2llnmeqat
StepHypRef Expression
1 simp3r 1024 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  .<_  ( X  ./\  Y
) )
2 hlatl 32342 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
323ad2ant1 1016 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  AtLat )
4 simp23 1030 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  e.  A )
5 simp1 995 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  HL )
6 simp21 1028 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  N )
7 simp22 1029 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  Y  e.  N )
8 simp3l 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  =/=  Y )
9 hllat 32345 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1093ad2ant1 1016 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  Lat )
11 eqid 2400 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
12 2llnmeqat.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1311, 12atbase 32271 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
144, 13syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  e.  ( Base `  K
) )
15 2llnmeqat.n . . . . . . . . 9  |-  N  =  ( LLines `  K )
1611, 15llnbase 32490 . . . . . . . 8  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
176, 16syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  ( Base `  K
) )
1811, 15llnbase 32490 . . . . . . . 8  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
197, 18syl 17 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  Y  e.  ( Base `  K
) )
20 2llnmeqat.l . . . . . . . 8  |-  .<_  =  ( le `  K )
21 2llnmeqat.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
2211, 20, 21latlem12 15922 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  X  /\  P  .<_  Y )  <-> 
P  .<_  ( X  ./\  Y ) ) )
2310, 14, 17, 19, 22syl13anc 1230 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  (
( P  .<_  X  /\  P  .<_  Y )  <->  P  .<_  ( X  ./\  Y )
) )
241, 23mpbird 232 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( P  .<_  X  /\  P  .<_  Y ) )
25 eqid 2400 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2620, 21, 25, 12, 152llnm4 32551 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  N  /\  Y  e.  N
)  /\  ( P  .<_  X  /\  P  .<_  Y ) )  ->  ( X  ./\  Y )  =/=  ( 0. `  K
) )
275, 4, 6, 7, 24, 26syl131anc 1241 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( X  ./\  Y )  =/=  ( 0. `  K
) )
2821, 25, 12, 152llnmat 32505 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  Y  e.  N )  /\  ( X  =/=  Y  /\  ( X  ./\  Y
)  =/=  ( 0.
`  K ) ) )  ->  ( X  ./\ 
Y )  e.  A
)
295, 6, 7, 8, 27, 28syl32anc 1236 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( X  ./\  Y )  e.  A )
3020, 12atcmp 32293 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( X  ./\  Y )  e.  A )  ->  ( P  .<_  ( X  ./\  Y )  <->  P  =  ( X  ./\  Y ) ) )
313, 4, 29, 30syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( P  .<_  ( X  ./\  Y )  <->  P  =  ( X  ./\  Y ) ) )
321, 31mpbid 210 1  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  =  ( X  ./\  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   meetcmee 15788   0.cp0 15881   Latclat 15889   Atomscatm 32245   AtLatcal 32246   HLchlt 32332   LLinesclln 32472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479
This theorem is referenced by:  cdlemeg46req  33512
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