Nuprl Lemma : midpoint-sep
∀e:BasicGeometry. ∀A,B,M:Point. (A ≠ B ⇒ A=M=B ⇒ {A ≠ M ∧ B ≠ M})
Proof
Definitions occuring in Statement :
geo-midpoint: a=m=b,
basic-geometry: BasicGeometry,
geo-sep: a ≠ b,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
member: t ∈ T,
cand: A c∧ B,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
guard: {T},
geo-midpoint: a=m=b
Lemmas referenced :
geo-point_wf,
geo-congruent_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-between_wf,
geo-congruent-symmetry,
geo-sep-sym,
geo-congruent-sep,
geo-sep_wf,
geo-sep-or
Rules used in proof :
instantiate,
productEquality,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
unionElimination,
because_Cache,
applyEquality,
isectElimination,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}A,B,M:Point. (A \mneq{} B {}\mRightarrow{} A=M=B {}\mRightarrow{} \{A \mneq{} M \mwedge{} B \mneq{} M\})
Date html generated:
2017_10_02-PM-06_34_12
Last ObjectModification:
2017_08_05-PM-04_44_16
Theory : euclidean!plane!geometry
Home
Index