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Theorem lplni 30018
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 961 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4179 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 3016 . . 3  |-  ( ( X  e.  N  /\  X C Y )  ->  E. x  e.  N  x C Y )
433ad2antl3 1121 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  E. x  e.  N  x C Y )
5 simpl1 960 . . 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 30016 . . 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 889 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2671   class class class wbr 4176   ` cfv 5417   Basecbs 13428    <o ccvr 29749   LLinesclln 29977   LPlanesclpl 29978
This theorem is referenced by:  lplnle  30026  llncvrlpln  30044  lplnexllnN  30050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-lplanes 29985
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