Nuprl Lemma : geo-line-from-points
∀e:EuclideanPlane. ∀a,b:Point. (a ≠ b ⇒ (∃l:Line. (a ≡ fst(l) ∧ b ≡ fst(snd(l)))))
Proof
Definitions occuring in Statement :
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
pi2: snd(t),
squash: ↓T,
sq_stable: SqStable(P),
pi1: fst(t),
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
geo-line: Line,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__geo-eq,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
subtype_rel_transitivity,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
exists_wf,
geo-sep_wf,
geo-eq_wf,
geo-eq_weakening
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
independent_pairFormation,
productEquality,
dependent_pairEquality,
productElimination,
lambdaEquality,
instantiate,
setEquality,
rename,
setElimination,
sqequalRule,
applyEquality,
dependent_set_memberEquality,
hypothesis,
independent_isectElimination,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
dependent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b:Point. (a \mneq{} b {}\mRightarrow{} (\mexists{}l:Line. (a \mequiv{} fst(l) \mwedge{} b \mequiv{} fst(snd(l)))))
Date html generated:
2018_05_22-PM-01_00_47
Last ObjectModification:
2018_01_16-PM-03_45_42
Theory : euclidean!plane!geometry
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