Nuprl Lemma : geo-strict-between-same-side
∀e:BasicGeometry. ∀[A,B,C,D:Point]. (¬((¬A-C-D) ∧ (¬A-D-C))) supposing (C ≠ D and A-B-C and A-B-D)
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q
Definitions unfolded in proof :
prop: ℙ,
and: P ∧ Q,
guard: {T},
subtype_rel: A ⊆r B,
false: False,
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
cand: A c∧ B,
geo-strict-between: a-b-c
Lemmas referenced :
geo-point_wf,
geo-sep_wf,
geo-strict-between_wf,
not_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-strict-between-sep2,
geo-strict-between-implies-between,
geo-between-same-side,
geo-between_wf,
geo-strict-between-sep1,
geo-sep-sym
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
lambdaEquality,
productEquality,
voidElimination,
productElimination,
independent_functionElimination,
sqequalRule,
instantiate,
applyEquality,
hypothesis,
because_Cache,
independent_isectElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation
Latex:
\mforall{}e:BasicGeometry. \mforall{}[A,B,C,D:Point]. (\mneg{}((\mneg{}A-C-D) \mwedge{} (\mneg{}A-D-C))) supposing (C \mneq{} D and A-B-C and A-B-D)
Date html generated:
2017_10_02-PM-06_17_27
Last ObjectModification:
2017_08_05-PM-04_12_35
Theory : euclidean!plane!geometry
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