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Theorem llni 34179
Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)
Hypotheses
Ref Expression
llnset.b  |-  B  =  ( Base `  K
)
llnset.c  |-  C  =  (  <o  `  K )
llnset.a  |-  A  =  ( Atoms `  K )
llnset.n  |-  N  =  ( LLines `  K )
Assertion
Ref Expression
llni  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  N )

Proof of Theorem llni
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 simpl2 995 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  B )
2 breq1 4443 . . . 4  |-  ( p  =  P  ->  (
p C X  <->  P C X ) )
32rspcev 3207 . . 3  |-  ( ( P  e.  A  /\  P C X )  ->  E. p  e.  A  p C X )
433ad2antl3 1155 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  E. p  e.  A  p C X )
5 simpl1 994 . . 3  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  K  e.  D )
6 llnset.b . . . 4  |-  B  =  ( Base `  K
)
7 llnset.c . . . 4  |-  C  =  (  <o  `  K )
8 llnset.a . . . 4  |-  A  =  ( Atoms `  K )
9 llnset.n . . . 4  |-  N  =  ( LLines `  K )
106, 7, 8, 9islln 34177 . . 3  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
115, 10syl 16 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
121, 4, 11mpbir2and 915 1  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   E.wrex 2808   class class class wbr 4440   ` cfv 5579   Basecbs 14479    <o ccvr 33934   Atomscatm 33935   LLinesclln 34162
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-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
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-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  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-iota 5542  df-fun 5581  df-fv 5587  df-llines 34169
This theorem is referenced by:  llnle  34189  atcvrlln  34191  lplncvrlvol  34287
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