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Theorem 2lplnmN 29748
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 960 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  P
)
2 simpl1 958 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  K  e.  HL )
3 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
4 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 2lplnm.p . . . . . 6  |-  P  =  ( LPlanes `  K )
64, 5lplnbase 29723 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
74, 5lplnbase 29723 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
8 2lplnm.m . . . . . 6  |-  ./\  =  ( meet `  K )
94, 8latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
103, 6, 7, 9syl3an 1224 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( X  ./\  Y
)  e.  ( Base `  K ) )
1110adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K ) )
1273ad2ant3 978 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  Y  e.  ( Base `  K ) )
1312adantr 451 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  (
Base `  K )
)
14 simp1 955 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  K  e.  HL )
1563ad2ant2 977 . . . . 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 29609 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) C Y  <->  X C
( X  .\/  Y
) ) )
1914, 15, 12, 18syl3anc 1182 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
2019biimpar 471 . . 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 29747 . . 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 1185 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( ( X 
./\  Y )  e.  N  <->  Y  e.  P
) )
241, 23mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   joincjn 14078   meetcmee 14079   Latclat 14151    <o ccvr 29452   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
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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