Nuprl Lemma : geo-extend-construction
∀e:EuclideanPlane. ∀q:Point. ∀a:{a:Point| q ≠ a} . ∀b,c:Point. (∃x:{Point| (q_a_x ∧ ax ≅ bc)})
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
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 :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
sq_exists: ∃x:{A| B[x]},
member: t ∈ T,
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
squash: ↓T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
euclidean-plane: EuclideanPlane
Lemmas referenced :
geo-sep_wf,
set_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
Euclid-Prop2-ext,
geo-congruent_wf,
geo-between_wf,
sq_stable__geo-sep,
extend-using-SC,
geo-congruent-transitivity
Rules used in proof :
lambdaEquality,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
isectElimination,
hypothesis,
rename,
setElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
productEquality,
independent_pairFormation,
dependent_set_memberFormation,
productElimination,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}q:Point. \mforall{}a:\{a:Point| q \mneq{} a\} . \mforall{}b,c:Point. (\mexists{}x:\{Point| (q\_a\_x \mwedge{} ax \00D0 bc)\})
Date html generated:
2017_10_02-PM-04_49_54
Last ObjectModification:
2017_08_06-PM-02_53_26
Theory : euclidean!plane!geometry
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