Nuprl Lemma : geo-colinear-from-between
∀e:BasicGeometry. ∀[A,C,D:Point]. (A ≠ C ⇒ (∃B:Point. (A ≠ B ∧ A_C_B ∧ A_D_B)) ⇒ Colinear(A;C;D))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
false: False,
not: ¬A,
geo-colinear: Colinear(a;b;c),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
not_wf,
geo-between_wf,
geo-sep_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
exists_wf,
euclidean-plane-axioms,
geo-colinear-between
Rules used in proof :
voidElimination,
isect_memberEquality,
because_Cache,
productEquality,
lambdaEquality,
sqequalRule,
instantiate,
applyEquality,
independent_functionElimination,
hypothesis,
independent_isectElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}[A,C,D:Point]. (A \mneq{} C {}\mRightarrow{} (\mexists{}B:Point. (A \mneq{} B \mwedge{} A\_C\_B \mwedge{} A\_D\_B)) {}\mRightarrow{} Colinear(A;C;D))
Date html generated:
2018_05_22-AM-11_57_02
Last ObjectModification:
2018_05_14-PM-03_18_06
Theory : euclidean!plane!geometry
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