Nuprl Lemma : triangle-inequality-for-colinear
∀[e:EuclideanPlane]. ∀[a,b,c:Point]. (Colinear(b;a;c) ⇒ |ac| ≤ |ab| + |bc|)
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-le: p ≤ q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-colinear: Colinear(a;b;c),
geo-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
basic-geometry-: BasicGeometry-,
all: ∀x:A. B[x],
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
geo-le: p ≤ q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
geo-simple-colinear-cases,
geo-le_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
stable__geo-le,
geo-add-length-between,
geo-between_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
subtype_rel_self,
iff_weakening_equal,
equal_wf,
geo-add-length-assoc,
geo-add-length-comm,
geo-le-add1,
geo-length-flip,
geo-le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
hypothesisEquality,
dependent_functionElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
independent_isectElimination,
applyEquality,
instantiate,
lambdaEquality,
imageElimination,
imageMemberEquality,
baseClosed,
isect_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[a,b,c:Point]. (Colinear(b;a;c) {}\mRightarrow{} |ac| \mleq{} |ab| + |bc|)
Date html generated:
2018_05_22-AM-11_56_41
Last ObjectModification:
2018_03_31-AM-00_56_04
Theory : euclidean!plane!geometry
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