Nuprl Lemma : at-most-one-midpoint
∀e:BasicGeometry. ∀P,P',A,B:Point. ((P=A=P' ∧ P=B=P') ⇒ A ≡ B)
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
false: False,
or: P ∨ Q,
not: ¬A,
uimplies: b supposing a,
stable: Stable{P},
geo-eq: a ≡ b,
and: P ∧ Q,
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
geo-midpoint: a=m=b,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q
Lemmas referenced :
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
geo-midpoint_wf,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
or_wf,
false_wf,
geo-sep_wf,
stable__not,
geo-length-flip,
geo-congruent-iff-length,
geo-sep-sym,
symmetric-point-construction,
geo-midpoint-symmetry,
geo-midpoint-diagonals-congruent,
geo-strict-between-congruence,
geo-sep-irrefl',
geo-midpoint-id2,
geo-sep_functionality,
geo-midpoint_functionality,
geo-congruent_functionality,
geo-eq_weakening,
geo-between_functionality
Rules used in proof :
instantiate,
productEquality,
voidElimination,
unionElimination,
independent_functionElimination,
independent_isectElimination,
productElimination,
lambdaFormation,
functionEquality,
sqequalRule,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
equalityTransitivity,
dependent_functionElimination,
rename,
equalitySymmetry,
promote_hyp
Latex:
\mforall{}e:BasicGeometry. \mforall{}P,P',A,B:Point. ((P=A=P' \mwedge{} P=B=P') {}\mRightarrow{} A \mequiv{} B)
Date html generated:
2017_10_02-PM-06_34_04
Last ObjectModification:
2017_08_05-PM-04_44_11
Theory : euclidean!plane!geometry
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