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Theorem lplni 29790
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 959 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4107 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 2960 . . 3  |-  ( ( X  e.  N  /\  X C Y )  ->  E. x  e.  N  x C Y )
433ad2antl3 1119 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  N
)  /\  X C Y )  ->  E. x  e.  N  x C Y )
5 simpl1 958 . . 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 29788 . . 3  |-  ( K  e.  D  ->  ( Y  e.  P  <->  ( Y  e.  B  /\  E. x  e.  N  x C Y ) ) )
115, 10syl 15 . 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 888 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 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   E.wrex 2620   class class class wbr 4104   ` cfv 5337   Basecbs 13245    <o ccvr 29521   LLinesclln 29749   LPlanesclpl 29750
This theorem is referenced by:  lplnle  29798  llncvrlpln  29816  lplnexllnN  29822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-lplanes 29757
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