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Theorem 2llnm2N 34239
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 1025 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  Y  e.  N )
2 simp1 991 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  HL )
3 hllat 34035 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
433ad2ant1 1012 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  K  e.  Lat )
5 simp21 1024 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  X  e.  N )
6 eqid 2460 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
7 2llnm2.n . . . . . 6  |-  N  =  ( LLines `  K )
86, 7llnbase 34180 . . . . 5  |-  ( X  e.  N  ->  X  e.  ( Base `  K
) )
95, 8syl 16 . . . 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 34180 . . . . 5  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
111, 10syl 16 . . . 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 15528 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
144, 9, 11, 13syl3anc 1223 . . 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 2460 . . . . . . 7  |-  ( join `  K )  =  (
join `  K )
17 2llnm2.p . . . . . . 7  |-  P  =  ( LPlanes `  K )
1815, 16, 7, 172llnjN 34238 . . . . . 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 1026 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  W  e.  P )
2018, 19eqeltrd 2548 . . . . 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 15536 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  X  .<_  ( X ( join `  K ) Y ) )
224, 9, 11, 21syl3anc 1223 . . . . 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 2460 . . . . . 6  |-  (  <o  `  K )  =  ( 
<o  `  K )
2415, 23, 7, 17llncvrlpln2 34228 . . . . 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 1226 . . . 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 34091 . . . . 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 1223 . . . 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 232 . . 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 34191 . . 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 1226 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521    <o ccvr 33934   Atomscatm 33935   HLchlt 34022   LLinesclln 34162   LPlanesclpl 34163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170
This theorem is referenced by:  2llnm3N  34240
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