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Theorem 2llnmeqat 29831
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 985 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  .<_  ( X  ./\  Y
) )
2 hlatl 29621 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
323ad2ant1 977 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  AtLat )
4 simp23 991 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  e.  A )
5 simp1 956 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  HL )
6 simp21 989 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  N )
7 simp22 990 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  Y  e.  N )
8 simp3l 984 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  =/=  Y )
9 hllat 29624 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1093ad2ant1 977 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  Lat )
11 eqid 2366 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
12 2llnmeqat.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1311, 12atbase 29550 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
144, 13syl 15 . . . . . . 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 29769 . . . . . . . 8  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
176, 16syl 15 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  ( Base `  K
) )
1811, 15llnbase 29769 . . . . . . . 8  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
197, 18syl 15 . . . . . . 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 14394 . . . . . . 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 1185 . . . . . 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 223 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( P  .<_  X  /\  P  .<_  Y ) )
25 eqid 2366 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2620, 21, 25, 12, 152llnm4 29830 . . . . 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 1196 . . . 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 29784 . . . 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 1191 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( X  ./\  Y )  e.  A )
3020, 12atcmp 29572 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( X  ./\  Y )  e.  A )  ->  ( P  .<_  ( X  ./\  Y )  <->  P  =  ( X  ./\  Y ) ) )
313, 4, 29, 30syl3anc 1183 . 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 201 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 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   meetcmee 14289   0.cp0 14353   Latclat 14361   Atomscatm 29524   AtLatcal 29525   HLchlt 29611   LLinesclln 29751
This theorem is referenced by:  cdlemeg46req  30789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758
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