Nuprl Lemma : eu-eq_dist-axiomsC-Pasch
∀e:EuclideanPlane. ∀x,y,z,u,t:Point.
(((((Dbet(e;x;t;u) ∧ Dsep(e;x;t)) ∧ Dbet(e;y;u;z)) ∧ Dtri(e;x;y;u)) ∧ Dtri(e;u;z;t))
⇒ (∃v:Point. ((Dbet(e;x;v;y) ∧ Dsep(e;x;v) ∧ Dsep(e;v;y)) ∧ Dbet(e;z;t;v) ∧ Dsep(e;z;t) ∧ Dsep(e;t;v))))
Proof
Definitions occuring in Statement :
dist-bet: Dbet(g;a;b;c),
dist-tri: Dtri(g;a;b;c),
dist-sep: Dsep(g;a;b),
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃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,
member: t ∈ T,
guard: {T},
cand: A c∧ B,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
geo-strict-between: a-b-c,
iff: P ⇐⇒ Q,
sq_exists: ∃x:A [B[x]],
euclidean-plane: EuclideanPlane,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
basic-geometry-: BasicGeometry-,
rev_implies: P ⇐ Q
Lemmas referenced :
lsep-all-sym,
Euclid-Prop20_cycle,
geo-lsep_wf,
outer-pasch-strict,
dist-bet_wf,
dist-sep_wf,
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,
Dbet-to-between,
geo-between-symmetry,
geo-between_wf,
Dsep-to-sep,
Dtri-iff-lsep,
lsep-implies-sep,
geo-sep-sym,
geo-strict-between_wf,
sq_stable__and,
sq_stable__geo-strict-between,
geo-strict-between-implies-between,
Dbet-iff-between,
Dsep-iff-sep
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
universeIsType,
isectElimination,
applyEquality,
sqequalRule,
productIsType,
inhabitedIsType,
instantiate,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
setElimination,
rename,
isect_memberEquality_alt,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}x,y,z,u,t:Point.
(((((Dbet(e;x;t;u) \mwedge{} Dsep(e;x;t)) \mwedge{} Dbet(e;y;u;z)) \mwedge{} Dtri(e;x;y;u)) \mwedge{} Dtri(e;u;z;t))
{}\mRightarrow{} (\mexists{}v:Point
((Dbet(e;x;v;y) \mwedge{} Dsep(e;x;v) \mwedge{} Dsep(e;v;y)) \mwedge{} Dbet(e;z;t;v) \mwedge{} Dsep(e;z;t) \mwedge{} Dsep(e;t;v))))
Date html generated:
2019_10_16-PM-02_58_39
Last ObjectModification:
2019_04_22-PM-02_21_45
Theory : euclidean!plane!geometry
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