Nuprl Lemma : Dtri-cycle
∀e:EuclideanPlane. ∀a,b,c:Point.
(Dtri(e;a;b;c) ⇒ {(Dtri(e;a;c;b) ∧ Dtri(e;c;a;b)) ∧ (Dtri(e;c;b;a) ∧ Dtri(e;b;a;c)) ∧ Dtri(e;b;c;a)})
Proof
Definitions occuring in Statement :
dist-tri: Dtri(g;a;b;c),
euclidean-plane: EuclideanPlane,
geo-point: Point,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev_implies: P ⇐ Q
Lemmas referenced :
Dtri-iff-lsep,
lsep-all-sym,
dist-tri_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
because_Cache,
independent_pairFormation,
universeIsType,
isectElimination,
inhabitedIsType,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c:Point.
(Dtri(e;a;b;c)
{}\mRightarrow{} \{(Dtri(e;a;c;b) \mwedge{} Dtri(e;c;a;b)) \mwedge{} (Dtri(e;c;b;a) \mwedge{} Dtri(e;b;a;c)) \mwedge{} Dtri(e;b;c;a)\})
Date html generated:
2019_10_16-PM-02_55_19
Last ObjectModification:
2019_02_27-AM-11_26_57
Theory : euclidean!plane!geometry
Home
Index