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Theorem 2llnmeqat 30099
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 29889 . . . 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 29892 . . . . . . . 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 2430 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
12 2llnmeqat.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1311, 12atbase 29818 . . . . . . . 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 30037 . . . . . . . 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 30037 . . . . . . . 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 14490 . . . . . . 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 2430 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
2620, 21, 25, 12, 152llnm4 30098 . . . . 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 30052 . . . 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 29840 . . 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 1652    e. wcel 1725    =/= wne 2593   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   Basecbs 13452   lecple 13519   meetcmee 14385   0.cp0 14449   Latclat 14457   Atomscatm 29792   AtLatcal 29793   HLchlt 29879   LLinesclln 30019
This theorem is referenced by:  cdlemeg46req  31057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-undef 6529  df-riota 6535  df-poset 14386  df-plt 14398  df-lub 14414  df-glb 14415  df-join 14416  df-meet 14417  df-p0 14451  df-lat 14458  df-clat 14520  df-oposet 29705  df-ol 29707  df-oml 29708  df-covers 29795  df-ats 29796  df-atl 29827  df-cvlat 29851  df-hlat 29880  df-llines 30026
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