Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnmeqat Structured version   Unicode version

Theorem 2llnmeqat 30442
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 987 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  .<_  ( X  ./\  Y
) )
2 hlatl 30232 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
323ad2ant1 979 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  AtLat )
4 simp23 993 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  P  e.  A )
5 simp1 958 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  HL )
6 simp21 991 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  e.  N )
7 simp22 992 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  Y  e.  N )
8 simp3l 986 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  X  =/=  Y )
9 hllat 30235 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1093ad2ant1 979 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  K  e.  Lat )
11 eqid 2438 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
12 2llnmeqat.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1311, 12atbase 30161 . . . . . . . 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 30380 . . . . . . . 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 30380 . . . . . . . 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 14512 . . . . . . 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 1187 . . . . . 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 225 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( P  .<_  X  /\  P  .<_  Y ) )
25 eqid 2438 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2620, 21, 25, 12, 152llnm4 30441 . . . . 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 1198 . . . 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 30395 . . . 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 1193 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  P  e.  A
)  /\  ( X  =/=  Y  /\  P  .<_  ( X  ./\  Y )
) )  ->  ( X  ./\  Y )  e.  A )
3020, 12atcmp 30183 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( X  ./\  Y )  e.  A )  ->  ( P  .<_  ( X  ./\  Y )  <->  P  =  ( X  ./\  Y ) ) )
313, 4, 29, 30syl3anc 1185 . 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 203 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   meetcmee 14407   0.cp0 14471   Latclat 14479   Atomscatm 30135   AtLatcal 30136   HLchlt 30222   LLinesclln 30362
This theorem is referenced by:  cdlemeg46req  31400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369
  Copyright terms: Public domain W3C validator