Nuprl Lemma : eu-middle_wf
∀[e:EuclideanStructure]. ∀[a,b:Point]. ∀[c:{c:Point| ¬Colinear(a;b;c)} ]. (middle(a;b;c) ∈ Point)
Proof
Definitions occuring in Statement :
eu-middle: middle(a;b;c),
eu-colinear: Colinear(a;b;c),
eu-point: Point,
euclidean-structure: EuclideanStructure,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eu-middle: middle(a;b;c),
eu-point: Point,
euclidean-structure: EuclideanStructure,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
prop: ℙ,
spreadn: spread3,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
eu-colinear: Colinear(a;b;c)
Lemmas referenced :
subtype_rel_self,
not_wf,
equal_wf,
uall_wf,
iff_wf,
isect_wf,
set_wf,
eu-point_wf,
eu-colinear_wf,
euclidean-structure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
sqequalHypSubstitution,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
hypothesis,
applyEquality,
tokenEquality,
instantiate,
extract_by_obid,
isectElimination,
universeEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
cumulativity,
hypothesisEquality,
because_Cache,
setEquality,
productEquality,
productElimination,
functionExtensionality,
lambdaFormation,
dependent_set_memberEquality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[e:EuclideanStructure]. \mforall{}[a,b:Point]. \mforall{}[c:\{c:Point| \mneg{}Colinear(a;b;c)\} ]. (middle(a;b;c) \mmember{} Point)
Date html generated:
2016_10_26-AM-07_40_46
Last ObjectModification:
2016_09_20-PM-08_00_47
Theory : euclidean!geometry
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