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Theorem 2llnmeqat 30065
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 986 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  .<_  ( X  ./\  Y
) )
2 hlatl 29855 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
323ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  AtLat )
4 simp23 992 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  e.  A )
5 simp1 957 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  HL )
6 simp21 990 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  N )
7 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  Y  e.  N )
8 simp3l 985 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  =/=  Y )
9 hllat 29858 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1093ad2ant1 978 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  Lat )
11 eqid 2412 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
12 2llnmeqat.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1311, 12atbase 29784 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
144, 13syl 16 . . . . . . 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 30003 . . . . . . . 8  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
176, 16syl 16 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  ( Base `  K
) )
1811, 15llnbase 30003 . . . . . . . 8  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
197, 18syl 16 . . . . . . 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 14470 . . . . . . 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 1186 . . . . . 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 224 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( P  .<_  X  /\  P  .<_  Y ) )
25 eqid 2412 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2620, 21, 25, 12, 152llnm4 30064 . . . . 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 1197 . . . 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 30018 . . . 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 1192 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( X  ./\  Y )  e.  A )
3020, 12atcmp 29806 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( X  ./\  Y )  e.  A )  ->  ( P  .<_  ( X  ./\  Y )  <->  P  =  ( X  ./\  Y ) ) )
313, 4, 29, 30syl3anc 1184 . 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 202 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   meetcmee 14365   0.cp0 14429   Latclat 14437   Atomscatm 29758   AtLatcal 29759   HLchlt 29845   LLinesclln 29985
This theorem is referenced by:  cdlemeg46req  31023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992
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