Nuprl Lemma : geo-between-implies-out
∀e:BasicGeometry. ∀p,a,b:Point. ((∃c:Point. (p ≠ c ∧ p_c_a ∧ p_c_b)) ⇒ out(p ab))
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
cand: A c∧ B,
basic-geometry: BasicGeometry,
geo-out: out(p ab),
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-between-sep,
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,
geo-between-same-side
Rules used in proof :
independent_functionElimination,
independent_pairFormation,
because_Cache,
productEquality,
lambdaEquality,
sqequalRule,
instantiate,
applyEquality,
hypothesis,
independent_isectElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}p,a,b:Point. ((\mexists{}c:Point. (p \mneq{} c \mwedge{} p\_c\_a \mwedge{} p\_c\_b)) {}\mRightarrow{} out(p ab))
Date html generated:
2017_10_02-PM-06_27_14
Last ObjectModification:
2017_10_02-PM-02_16_04
Theory : euclidean!plane!geometry
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