Nuprl Lemma : Dbet-to-le
∀g:EuclideanPlane. ∀a,b,c:Point. (Dbet(g;a;b;c) ⇒ |ab| + |bc| ≤ |ac|)
Proof
Definitions occuring in Statement :
dist-bet: Dbet(g;a;b;c),
geo-le: p ≤ q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
not: ¬A,
dist-bet: Dbet(g;a;b;c),
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
not-dist-to-le,
dist-bet_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
hypothesisEquality,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c:Point. (Dbet(g;a;b;c) {}\mRightarrow{} |ab| + |bc| \mleq{} |ac|)
Date html generated:
2019_10_16-PM-02_53_24
Last ObjectModification:
2018_12_04-PM-02_24_26
Theory : euclidean!plane!geometry
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