Nuprl Lemma : dist-iff-lt
∀g:EuclideanPlane. ∀a,b,c,d,e,f:Point. (|ef| < |ab| + |cd| ⇐⇒ D(a;b;c;d;e;f))
Proof
Definitions occuring in Statement :
dist: D(a;b;c;d;e;f),
geo-lt: 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],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
dist-lemma-lt2,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
dist-lemma-lt,
dist_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,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
universeIsType,
isectElimination,
sqequalRule,
setElimination,
rename,
because_Cache,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point. (|ef| < |ab| + |cd| \mLeftarrow{}{}\mRightarrow{} D(a;b;c;d;e;f))
Date html generated:
2019_10_16-PM-02_51_02
Last ObjectModification:
2018_10_03-AM-11_24_39
Theory : euclidean!plane!geometry
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