Nuprl Lemma : geo-strict-between-incident
∀[e:EuclideanPlane]. ∀[l:LINE]. ∀[a,b,v:Point]. (a-v-b ⇒ a I l ⇒ b I l ⇒ v I l)
Proof
Definitions occuring in Statement :
geo-incident: p I L,
geoline: LINE,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-point: Point,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
basic-geometry-: BasicGeometry-,
basic-geometry: BasicGeometry,
uimplies: b supposing a,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
geo-incident: p I L,
geo-colinear: Colinear(a;b;c),
subtype_rel: A ⊆r B,
guard: {T},
geo-line: Line,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
geo-colinear-incident,
geo-strict-between-sep1,
geo-colinear-is-colinear-set,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
false_wf,
lelt_wf,
not_wf,
geo-between_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
equal_wf,
geoline_wf,
geoline-subtype1,
geo-line_wf,
geo-incident_wf,
geo-strict-between_wf,
geo-point_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaEquality,
productEquality,
applyEquality,
instantiate,
productElimination
Latex:
\mforall{}[e:EuclideanPlane]. \mforall{}[l:LINE]. \mforall{}[a,b,v:Point]. (a-v-b {}\mRightarrow{} a I l {}\mRightarrow{} b I l {}\mRightarrow{} v I l)
Date html generated:
2018_05_22-PM-01_04_50
Last ObjectModification:
2018_05_11-PM-11_11_45
Theory : euclidean!plane!geometry
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