Nuprl Lemma : geo-lt-from-strict-between
∀e:EuclideanPlane. ∀a,b,c:Point. (a-b-c ⇒ |ab| < |ac|)
Proof
Definitions occuring in Statement :
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,
geo-strict-between: a-b-c,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
euclidean-plane: EuclideanPlane,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
geo-lt: p < q,
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
geo-add-length-between,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-length_wf,
geo-mk-seg_wf,
subtype_rel_self,
iff_weakening_equal,
geo-le-same,
geo-sep_wf,
geo-le_wf,
geo-add-length_wf,
geo-strict-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
because_Cache,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point. (a-b-c {}\mRightarrow{} |ab| < |ac|)
Date html generated:
2019_10_16-PM-01_20_24
Last ObjectModification:
2019_07_24-PM-00_46_39
Theory : euclidean!plane!geometry
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