Nuprl Lemma : dist-lemma-lt2
∀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],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
dist: D(a;b;c;d;e;f),
exists: ∃x:A. B[x],
member: t ∈ T,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
iff: P ⇐⇒ Q
Lemmas referenced :
geo-X_wf,
geo-length_wf1,
geo-mk-seg_wf,
geo-add-length_wf1,
geo-add-length-property1,
geo-length-property,
geo-add-length-property2,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent_wf,
geo-sep_wf,
geo-lt_wf,
geo-length_wf,
geo-add-length_wf,
geo-point_wf,
geo-lt-iff-strict-between-points,
geo-sep-sym,
geo-le-iff-between-points
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
isectElimination,
sqequalRule,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
productIsType,
universeIsType,
instantiate,
independent_isectElimination,
setIsType,
productElimination,
independent_functionElimination,
dependent_set_memberEquality_alt
Latex:
\mforall{}g:EuclideanPlane. \mforall{}a,b,c,d,e,f:Point. (|ef| < |ab| + |cd| {}\mRightarrow{} D(a;b;c;d;e;f))
Date html generated:
2019_10_16-PM-02_50_44
Last ObjectModification:
2018_10_03-AM-11_23_23
Theory : euclidean!plane!geometry
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