Nuprl Lemma : Euclid-parallel-exists
∀e:EuclideanPlane. ∀l:Line. ∀p:Point. ∃m:Line. (m || l ∧ p I m)
Proof
Definitions occuring in Statement :
geo-Aparallel: l || m,
geo-incident: p I L,
geo-line: Line,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
geo-line: Line,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
basic-geometry: BasicGeometry,
guard: {T},
uimplies: b supposing a,
euclidean-plane: EuclideanPlane,
cand: A c∧ B,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
geo-Aparallel: l || m,
not: ¬A,
or: P ∨ Q,
false: False,
stable: Stable{P},
geo-eq: a ≡ b,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
geoline: LINE,
quotient: x,y:A//B[x; y],
pi1: fst(t),
pi2: snd(t),
geo-perp-in: ab ⊥x cd,
oriented-plane: OrientedPlane,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
select: L[n],
cons: [a / b],
subtract: n - m,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
true: True,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
Euclid-drop-perp-0,
geo-sep_wf,
Euclid-erect-perp,
geo-colinear-same,
geo-colinear_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-line_wf,
geo-perp-in_wf,
geo-lsep_wf,
sq_stable__and,
sq_stable__colinear,
sq_stable__geo-perp-in,
sq_stable__geo-lsep,
sq_stable__geo-sep,
geo-sep-sym,
lsep-implies-sep,
geo-Aparallel_wf,
geoline-subtype1,
geo-incident_wf,
geo-intersect_wf,
stable__false,
false_wf,
not_wf,
istype-void,
minimal-double-negation-hyp-elim,
geo-perp-in_functionality,
geo-eq_weakening,
geo-colinear_functionality,
minimal-not-not-excluded-middle,
geo-intersect-iff3,
geo-line-eq-to-col,
geo-line-eq_wf,
right-angles-not-complementary,
oriented-colinear-append,
cons_wf,
nil_wf,
cons_member,
l_member_wf,
geo-colinear-is-colinear-set,
list_ind_cons_lemma,
list_ind_nil_lemma,
geo-strict-between-implies-colinear,
length_of_cons_lemma,
length_of_nil_lemma,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
right-angle-symmetry,
adjacent-right-angles,
euclidean-plane-axioms,
quotient-member-eq,
geo-line-eq-equiv,
geo-colinear-line-eq2,
geo-intersect-irreflexive,
squash_wf,
true_wf,
geoline_wf,
subtype_rel_self,
iff_weakening_equal,
geo-incident-line
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
sqequalRule,
rename,
setElimination,
instantiate,
independent_isectElimination,
isect_memberEquality_alt,
productEquality,
independent_pairFormation,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation_alt,
dependent_pairEquality_alt,
productIsType,
inhabitedIsType,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
unionElimination,
voidElimination,
promote_hyp,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inrFormation_alt,
inlFormation_alt,
natural_numberEquality,
approximateComputation,
lambdaEquality_alt,
universeEquality,
independent_pairEquality
Latex:
\mforall{}e:EuclideanPlane. \mforall{}l:Line. \mforall{}p:Point. \mexists{}m:Line. (m || l \mwedge{} p I m)
Date html generated:
2019_10_16-PM-02_42_13
Last ObjectModification:
2018_12_15-PM-09_45_47
Theory : euclidean!plane!geometry
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