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Theorem 2lplnmN 30418
Description: If the join of two lattice planes covers one of them, their meet is a lattice line. (Contributed by NM, 30-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2lplnm.j  |-  .\/  =  ( join `  K )
2lplnm.m  |-  ./\  =  ( meet `  K )
2lplnm.c  |-  C  =  (  <o  `  K )
2lplnm.n  |-  N  =  ( LLines `  K )
2lplnm.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
2lplnmN  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  N )

Proof of Theorem 2lplnmN
StepHypRef Expression
1 simpl3 963 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  P
)
2 simpl1 961 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  K  e.  HL )
3 hllat 30223 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
4 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 2lplnm.p . . . . . 6  |-  P  =  ( LPlanes `  K )
64, 5lplnbase 30393 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
74, 5lplnbase 30393 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
8 2lplnm.m . . . . . 6  |-  ./\  =  ( meet `  K )
94, 8latmcl 14482 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
103, 6, 7, 9syl3an 1227 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( X  ./\  Y
)  e.  ( Base `  K ) )
1110adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K ) )
1273ad2ant3 981 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  Y  e.  ( Base `  K ) )
1312adantr 453 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  (
Base `  K )
)
14 simp1 958 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  K  e.  HL )
1563ad2ant2 980 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  X  e.  ( Base `  K ) )
16 2lplnm.j . . . . . 6  |-  .\/  =  ( join `  K )
17 2lplnm.c . . . . . 6  |-  C  =  (  <o  `  K )
184, 16, 8, 17cvrexch 30279 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) C Y  <->  X C
( X  .\/  Y
) ) )
1914, 15, 12, 18syl3anc 1185 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
2019biimpar 473 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y ) C Y )
21 2lplnm.n . . . 4  |-  N  =  ( LLines `  K )
224, 17, 21, 5llncvrlpln 30417 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  ./\  Y
)  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  /\  ( X  ./\  Y ) C Y )  ->  (
( X  ./\  Y
)  e.  N  <->  Y  e.  P ) )
232, 11, 13, 20, 22syl31anc 1188 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( ( X 
./\  Y )  e.  N  <->  Y  e.  P
) )
241, 23mpbird 225 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   joincjn 14403   meetcmee 14404   Latclat 14476    <o ccvr 30122   HLchlt 30210   LLinesclln 30350   LPlanesclpl 30351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358
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