Nuprl Lemma : eu-exists-middle
∀e:EuclideanPlane. ∀a,b:Point. ∀c:{p:Point| ¬Colinear(a;b;p)} . ∃m:Point. ((m = middle(a;b;c) ∈ Point) ∧ am=bm ∧ am=cm)
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
eu-middle: middle(a;b;c),
eu-colinear: Colinear(a;b;c),
eu-congruent: ab=cd,
eu-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
euclidean-plane: EuclideanPlane,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
euclidean-axioms: euclidean-axioms(e),
and: P ∧ Q,
cand: A c∧ B,
let: let
Lemmas referenced :
sq_stable__eu-congruent,
sq_stable__equal,
sq_stable__and,
eu-congruent_wf,
equal_wf,
and_wf,
eu-middle_wf,
euclidean-plane_wf,
eu-colinear_wf,
not_wf,
eu-point_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_pairFormation,
isect_memberEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
independent_pairFormation
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. \mforall{}c:\{p:Point| \mneg{}Colinear(a;b;p)\} .
\mexists{}m:Point. ((m = middle(a;b;c)) \mwedge{} am=bm \mwedge{} am=cm)
Date html generated:
2016_05_18-AM-06_35_41
Last ObjectModification:
2016_01_16-PM-10_30_46
Theory : euclidean!geometry
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