Nuprl Lemma : Dsep-to-sep
∀e:EuclideanPlane. ∀a,b:Point. (Dsep(e;a;b) ⇒ a ≠ b)
Proof
Definitions occuring in Statement :
dist-sep: Dsep(g;a;b),
euclidean-plane: EuclideanPlane,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dist-sep: Dsep(g;a;b),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
squash: ↓T,
euclidean-plane: EuclideanPlane,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
dist-sep_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length-zero2,
subtype_rel_self,
iff_weakening_equal,
dist-iff-lt,
geo-lt-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. (Dsep(e;a;b) {}\mRightarrow{} a \mneq{} b)
Date html generated:
2019_10_16-PM-02_54_23
Last ObjectModification:
2019_02_18-AM-08_22_38
Theory : euclidean!plane!geometry
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