Nuprl Lemma : geo-lt-out-to-between
∀e:EuclideanPlane. ∀a,b,c:Point. (out(a bc) ⇒ |ab| < |ac| ⇒ a-b-c)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
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,
basic-geometry: BasicGeometry,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
euclidean-plane: EuclideanPlane,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
geo-strict-between: a-b-c,
cand: A c∧ B,
geo-out: out(p ab),
squash: ↓T,
true: True
Lemmas referenced :
geo-out-le-iff-bet,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-out_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-le_weakening-lt,
geo-add-length-between,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-cancel-left-lt2,
geo-zero-lt-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
universeIsType,
isectElimination,
setElimination,
rename,
because_Cache,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
independent_pairFormation,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (out(a bc) {}\mRightarrow{} |ab| < |ac| {}\mRightarrow{} a-b-c)
Date html generated:
2019_10_16-PM-01_24_09
Last ObjectModification:
2018_12_13-PM-10_32_35
Theory : euclidean!plane!geometry
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