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Theorem 2lplnmN 34232
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 996 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  P
)
2 simpl1 994 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  K  e.  HL )
3 hllat 34037 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
4 eqid 2462 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 2lplnm.p . . . . . 6  |-  P  =  ( LPlanes `  K )
64, 5lplnbase 34207 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
74, 5lplnbase 34207 . . . . 5  |-  ( Y  e.  P  ->  Y  e.  ( Base `  K
) )
8 2lplnm.m . . . . . 6  |-  ./\  =  ( meet `  K )
94, 8latmcl 15530 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  ( X  ./\  Y )  e.  ( Base `  K
) )
103, 6, 7, 9syl3an 1265 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( X  ./\  Y
)  e.  ( Base `  K ) )
1110adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  ( X  ./\  Y )  e.  ( Base `  K ) )
1273ad2ant3 1014 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  Y  e.  ( Base `  K ) )
1312adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  /\  X C ( X 
.\/  Y ) )  ->  Y  e.  (
Base `  K )
)
14 simp1 991 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  K  e.  HL )
1563ad2ant2 1013 . . . . 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 34093 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K
) )  ->  (
( X  ./\  Y
) C Y  <->  X C
( X  .\/  Y
) ) )
1914, 15, 12, 18syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  P )  ->  ( ( X  ./\  Y ) C Y  <->  X C
( X  .\/  Y
) ) )
2019biimpar 485 . . 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 34231 . . 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 1226 . 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   joincjn 15422   meetcmee 15423   Latclat 15523    <o ccvr 33936   HLchlt 34024   LLinesclln 34164   LPlanesclpl 34165
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172
This theorem is referenced by: (None)
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