Nuprl Lemma : in-hull-unique3
∀[g:OrientedPlane]. ∀[xs:{xs:Point List| geo-general-position(g;xs)} ]. ∀[i,j,k:ℕ||xs||].
((((((¬(j = k ∈ ℤ)) ∧ (¬(j = i ∈ ℤ))) ∧ (¬(k = i ∈ ℤ))) ∧ jk ∈ Hull(xs)) ∧ ji ∈ Hull(xs)) ⇒ False)
Proof
Definitions occuring in Statement :
in-hull: ij ∈ Hull(xs),
geo-general-position: geo-general-position(g;xs),
oriented-plane: OrientedPlane,
geo-point: Point,
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
false: False,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
prop: ℙ,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
and: P ∧ Q
Lemmas referenced :
list_wf,
set_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
oriented-plane_wf,
subtype_rel_transitivity,
oriented-plane-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
length_wf,
int_seg_wf,
geo-general-position_wf,
in-hull_wf,
equal_wf,
not_wf,
in-hull-unique1
Rules used in proof :
lambdaEquality,
sqequalRule,
instantiate,
applyEquality,
natural_numberEquality,
dependent_set_memberEquality,
rename,
setElimination,
intEquality,
productEquality,
voidElimination,
independent_functionElimination,
hypothesis,
because_Cache,
independent_isectElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
isect_memberFormation,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality
Latex:
\mforall{}[g:OrientedPlane]. \mforall{}[xs:\{xs:Point List| geo-general-position(g;xs)\} ]. \mforall{}[i,j,k:\mBbbN{}||xs||].
((((((\mneg{}(j = k)) \mwedge{} (\mneg{}(j = i))) \mwedge{} (\mneg{}(k = i))) \mwedge{} jk \mmember{} Hull(xs)) \mwedge{} ji \mmember{} Hull(xs)) {}\mRightarrow{} False)
Date html generated:
2018_05_22-PM-00_07_18
Last ObjectModification:
2017_12_12-PM-03_05_32
Theory : euclidean!plane!geometry
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