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Theorem llni 32505
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 1001 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  X  e.  B )
2 breq1 4397 . . . 4  |-  ( p  =  P  ->  (
p C X  <->  P C X ) )
32rspcev 3159 . . 3  |-  ( ( P  e.  A  /\  P C X )  ->  E. p  e.  A  p C X )
433ad2antl3 1161 . 2  |-  ( ( ( K  e.  D  /\  X  e.  B  /\  P  e.  A
)  /\  P C X )  ->  E. p  e.  A  p C X )
5 simpl1 1000 . . 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 32503 . . 3  |-  ( K  e.  D  ->  ( X  e.  N  <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
115, 10syl 17 . 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 923 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   E.wrex 2754   class class class wbr 4394   ` cfv 5568   Basecbs 14839    <o ccvr 32260   Atomscatm 32261   LLinesclln 32488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-llines 32495
This theorem is referenced by:  llnle  32515  atcvrlln  32517  lplncvrlvol  32613
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