Nuprl Lemma : canonical-parallel
∀e:EuclideanParPlane. ∀c:l,m:Line//l || m. (∃l:LINE [(l = c ∈ (l,m:Line//l || m))])
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-Aparallel: l || m,
geoline: LINE,
geo-line: Line,
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
euclidean-parallel-plane: EuclideanParPlane,
so_apply: x[s1;s2],
uimplies: b supposing a,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
quotient: x,y:A//B[x; y],
euclidean-plane: EuclideanPlane,
so_lambda: λ2x.t[x],
so_apply: x[s],
geoline: LINE,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
quotient_wf,
geo-line_wf,
geo-Aparallel_wf,
geo-Aparallel-equiv,
euclidean-parallel-plane_wf,
Euclid-parallel-exists,
geo-Aparallel_inversion,
geoline-subtype1,
geo-incident_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
euclidean-planes-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-point_wf,
geoline_wf,
geo-O_wf,
subtype_rel_self,
sq_exists_wf,
geo-playfair-axiom,
geo-Aparallel_transitivity,
geo-Aparallel_weakening2,
quotient-member-eq,
geo-line-eq_wf,
geo-line-eq-equiv,
sq_stable__geo-Aparallel,
geoline-subtype2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
independent_isectElimination,
productElimination,
dependent_set_memberEquality_alt,
independent_functionElimination,
independent_pairFormation,
productIsType,
instantiate,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
equalityIstype,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
functionEquality,
productEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}e:EuclideanParPlane. \mforall{}c:l,m:Line//l || m. (\mexists{}l:LINE [(l = c)])
Date html generated:
2019_10_16-PM-02_43_32
Last ObjectModification:
2018_12_11-PM-11_43_40
Theory : euclidean!plane!geometry
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