Nuprl Lemma : Euclid-Prop20_cycle
∀e:EuclideanPlane. ∀a,b,c:Point. (a # bc ⇒ (|bc| < |ba| + |ac| ∧ |ac| < |ba| + |bc| ∧ |ba| < |ac| + |bc|))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-add-length: p + q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-lsep: a # bc,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
Euclid-Prop20,
geo-lsep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
lsep-all-sym,
geo-lt_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-add-length_wf,
geo-length-flip,
subtype_rel_self,
iff_weakening_equal,
geo-add-length-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
universeIsType,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
because_Cache,
productElimination,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
(a \# bc {}\mRightarrow{} (|bc| < |ba| + |ac| \mwedge{} |ac| < |ba| + |bc| \mwedge{} |ba| < |ac| + |bc|))
Date html generated:
2019_10_16-PM-02_19_46
Last ObjectModification:
2019_02_17-PM-00_35_09
Theory : euclidean!plane!geometry
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