Nuprl Lemma : geo-midpoint-implies-between
∀e:BasicGeometry. ∀a,b,m:Point. (a ≠ b ⇒ a=m=b ⇒ a-m-b)
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-strict-between: a-b-c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
and: P ∧ Q,
exists: ∃x:A. B[x],
geo-midpoint: a=m=b,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q
Lemmas referenced :
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
geo-midpoint_wf,
midpoint-sep,
geo-extend-exists1,
geo-sep-sym,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-congruent-iff-length,
geo-congruent-symmetry,
symmetric-point-unicity,
geo-strict-between_functionality,
geo-eq_weakening,
geo-midpoint_functionality
Rules used in proof :
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productElimination,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,m:Point. (a \mneq{} b {}\mRightarrow{} a=m=b {}\mRightarrow{} a-m-b)
Date html generated:
2017_10_02-PM-06_44_12
Last ObjectModification:
2017_08_05-PM-04_49_58
Theory : euclidean!plane!geometry
Home
Index