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Theorem lplni 30427
Description: Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.)
Hypotheses
Ref Expression
lplnset.b  |-  B  =  ( Base `  K
)
lplnset.c  |-  C  =  (  <o  `  K )
lplnset.n  |-  N  =  ( LLines `  K )
lplnset.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplni  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  Y  e.  P )

Proof of Theorem lplni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4240 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 3058 . . 3  |-  ( ( X  e.  N  /\  X C Y )  ->  E. x  e.  N  x C Y )
433ad2antl3 1122 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  E. x  e.  N  x C Y )
5 simpl1 961 . . 3  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  K  e.  D )
6 lplnset.b . . . 4  |-  B  =  ( Base `  K
)
7 lplnset.c . . . 4  |-  C  =  (  <o  `  K )
8 lplnset.n . . . 4  |-  N  =  ( LLines `  K )
9 lplnset.p . . . 4  |-  P  =  ( LPlanes `  K )
106, 7, 8, 9islpln 30425 . . 3  |-  ( K  e.  D  ->  ( Y  e.  P  <->  ( Y  e.  B  /\  E. x  e.  N  x C Y ) ) )
115, 10syl 16 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  ( Y  e.  P  <->  ( Y  e.  B  /\  E. x  e.  N  x C Y ) ) )
121, 4, 11mpbir2and 890 1  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  Y  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   E.wrex 2712   class class class wbr 4237   ` cfv 5483   Basecbs 13500    <o ccvr 30158   LLinesclln 30386   LPlanesclpl 30387
This theorem is referenced by:  lplnle  30435  llncvrlpln  30453  lplnexllnN  30459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-lplanes 30394
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