Nuprl Lemma : Euclid-Prop10
∀e:EuclideanPlane. ∀a:Point. ∀b:{b:Point| a ≠ b} . (∃d:{Point| (a-d-b ∧ ad ≅ db)})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
set: {x:A| B[x]}
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
geo-midpoint: a=m=b,
geo-strict-between: a-b-c,
implies: P ⇒ Q,
euclidean-plane: EuclideanPlane,
or: P ∨ Q
Lemmas referenced :
Euclid-midpoint,
geo-strict-between_wf,
geo-congruent_wf,
set_wf,
geo-point_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-congruent-symmetry,
geo-congruent-sep,
geo-sep-or,
geo-sep-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
hypothesis,
productEquality,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
instantiate,
independent_isectElimination,
lambdaEquality,
productElimination,
independent_functionElimination,
unionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a:Point. \mforall{}b:\{b:Point| a \mneq{} b\} . (\mexists{}d:\{Point| (a-d-b \mwedge{} ad \00D0 db)\})
Date html generated:
2017_10_02-PM-06_55_11
Last ObjectModification:
2017_08_16-AM-00_59_52
Theory : euclidean!plane!geometry
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