Nuprl Lemma : symmetric-point-construction
∀e:BasicGeometry. ∀a,p:Point. (a ≠ p ⇒ (∃p':Point. p=a=p'))
Proof
Definitions occuring in Statement :
basic-geometry: BasicGeometry,
geo-midpoint: a=m=b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sympoint_wf,
geo-sep_wf,
geo-midpoint_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
sq_stable__midpoint
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIsType1
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,p:Point. (a \mneq{} p {}\mRightarrow{} (\mexists{}p':Point. p=a=p'))
Date html generated:
2019_10_16-PM-01_15_09
Last ObjectModification:
2018_11_08-PM-01_16_32
Theory : euclidean!plane!geometry
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