Nuprl Lemma : geo-extend-exists
∀e:EuclideanPlane. ∀q,a,b,c:Point. (q ≠ a ⇒ (∃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],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
euclidean-plane: EuclideanPlane,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
uimplies: b supposing a,
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__geo-congruent,
sq_stable__geo-between,
sq_stable__and,
equal_wf,
set_wf,
geo-congruent_wf,
geo-between_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
geo-sep_wf,
geo-extend_wf
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
isect_memberEquality,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
productEquality,
independent_isectElimination,
instantiate,
setEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
applyEquality,
isectElimination,
hypothesis,
because_Cache,
dependent_set_memberEquality,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
dependent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}q,a,b,c:Point. (q \mneq{} a {}\mRightarrow{} (\mexists{}x:Point. (q\_a\_x \mwedge{} ax \00D0 bc)))
Date html generated:
2017_10_02-PM-04_50_33
Last ObjectModification:
2017_08_06-PM-03_00_56
Theory : euclidean!plane!geometry
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