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Theorem 2llnm2N 29757
Description: The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2llnm2.l  |-  .<_  =  ( le `  K )
2llnm2.m  |-  ./\  =  ( meet `  K )
2llnm2.a  |-  A  =  ( Atoms `  K )
2llnm2.n  |-  N  =  ( LLines `  K )
2llnm2.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
2llnm2N  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  A )

Proof of Theorem 2llnm2N
StepHypRef Expression
1 simp22 989 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  N )
2 simp1 955 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  HL )
3 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  Lat )
5 simp21 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  N )
6 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2llnm2.n . . . . . 6  |-  N  =  ( LLines `  K )
86, 7llnbase 29698 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
95, 8syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  ( Base `  K
) )
106, 7llnbase 29698 . . . . 5  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
111, 10syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  ( Base `  K
) )
12 2llnm2.m . . . . 5  |-  ./\  =  ( meet `  K )
136, 12latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
15 2llnm2.l . . . . . . 7  |-  .<_  =  ( le `  K )
16 eqid 2283 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
17 2llnm2.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
1815, 16, 7, 172llnjN 29756 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X ( join `  K
) Y )  =  W )
19 simp23 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  W  e.  P )
2018, 19eqeltrd 2357 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X ( join `  K
) Y )  e.  P )
216, 15, 16latlej1 14166 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
224, 9, 11, 21syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
23 eqid 2283 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
2415, 23, 7, 17llncvrlpln2 29746 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  N  /\  ( X ( join `  K
) Y )  e.  P )  /\  X  .<_  ( X ( join `  K ) Y ) )  ->  X (  <o  `  K ) ( X ( join `  K
) Y ) )
252, 5, 20, 22, 24syl31anc 1185 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X
(  <o  `  K )
( X ( join `  K ) Y ) )
266, 16, 12, 23cvrexch 29609 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) (  <o  `  K
) Y  <->  X (  <o  `  K ) ( X ( join `  K
) Y ) ) )
272, 9, 11, 26syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( X  ./\  Y
) (  <o  `  K
) Y  <->  X (  <o  `  K ) ( X ( join `  K
) Y ) ) )
2825, 27mpbird 223 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y ) ( 
<o  `  K ) Y )
29 2llnm2.a . . . 4  |-  A  =  ( Atoms `  K )
306, 23, 29, 7atcvrlln 29709 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  /\  ( X  ./\  Y ) ( 
<o  `  K ) Y )  ->  ( ( X  ./\  Y )  e.  A  <->  Y  e.  N
) )
312, 14, 11, 28, 30syl31anc 1185 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( X  ./\  Y
)  e.  A  <->  Y  e.  N ) )
321, 31mpbird 223 1  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  ./\  Y )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  2llnm3N  29758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
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