Nuprl Lemma : Dbet-iff-between
∀e:EuclideanPlane. ∀a,b,c:Point. ((∀A,B,C:Point. (A # BC ⇒ |AC| < |AB| + |BC|)) ⇒ (Dbet(e;a;b;c) ⇐⇒ a_b_c))
Proof
Definitions occuring in Statement :
dist-bet: Dbet(g;a;b;c),
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-between: a_b_c,
geo-point: Point,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
dist-bet: Dbet(g;a;b;c),
not: ¬A,
squash: ↓T,
true: True,
false: False
Lemmas referenced :
dist-bet_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lsep_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-point_wf,
Dbet-to-between,
dist_wf,
geo-add-length-between,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
subtype_rel_self,
iff_weakening_equal,
geo-lt-irrefl,
dist-iff-lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
functionIsType,
inhabitedIsType,
because_Cache,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
((\mforall{}A,B,C:Point. (A \# BC {}\mRightarrow{} |AC| < |AB| + |BC|)) {}\mRightarrow{} (Dbet(e;a;b;c) \mLeftarrow{}{}\mRightarrow{} a\_b\_c))
Date html generated:
2019_10_16-PM-02_54_05
Last ObjectModification:
2018_12_05-PM-03_05_24
Theory : euclidean!plane!geometry
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