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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