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Theorem 2lplnmN 32540
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 1000 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  P
)
2 simpl1 998 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  K  e.  HL )
3 hllat 32345 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
4 eqid 2400 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 2lplnm.p . . . . . 6  |-  P  =  ( LPlanes `  K )
64, 5lplnbase 32515 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
74, 5lplnbase 32515 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
8 2lplnm.m . . . . . 6  |-  ./\  =  ( meet `  K )
94, 8latmcl 15896 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
103, 6, 7, 9syl3an 1270 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( X  ./\  Y
)  e.  ( Base `  K ) )
1110adantr 463 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K ) )
1273ad2ant3 1018 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  Y  e.  ( Base `  K ) )
1312adantr 463 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  (
Base `  K )
)
14 simp1 995 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  K  e.  HL )
1563ad2ant2 1017 . . . . 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 32401 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) C Y  <->  X C
( X  .\/  Y
) ) )
1914, 15, 12, 18syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
2019biimpar 483 . . 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 32539 . . 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 1231 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( ( X 
./\  Y )  e.  N  <->  Y  e.  P
) )
241, 23mpbird 232 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   joincjn 15787   meetcmee 15788   Latclat 15889    <o ccvr 32244   HLchlt 32332   LLinesclln 32472   LPlanesclpl 32473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-llines 32479  df-lplanes 32480
This theorem is referenced by: (None)
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