Nuprl Lemma : geo-intersect-unique
∀eu:EuclideanParPlane. ∀l,m:Line. ∀x,y:Point. (l \/ m ⇒ (x I l ∧ x I m) ⇒ (y I l ∧ y I m) ⇒ x ≡ y)
Proof
Definitions occuring in Statement :
euclidean-parallel-plane: EuclideanParPlane,
geo-intersect: L \/ M,
geo-incident: p I L,
geo-line: Line,
geo-eq: a ≡ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
euclidean-parallel-plane: EuclideanParPlane,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
geo-line-sep: (l # m),
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
geo-plsep: p # l,
geo-lsep: a # bc,
or: P ∨ Q,
pi2: snd(t),
pi1: fst(t),
top: Top,
geo-line: Line,
euclidean-plane: EuclideanPlane,
subtract: n - m,
cons: [a / b],
select: L[n],
true: True,
squash: ↓T,
less_than: a < b,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_all: (∀x∈L.P[x]),
geo-colinear-set: geo-colinear-set(e; L),
rev_implies: P ⇐ Q,
not: ¬A
Lemmas referenced :
geo-line_wf,
geo-primitives_wf,
euclidean-plane-structure_wf,
euclidean-plane_wf,
euclidean-parallel-plane_wf,
subtype_rel_transitivity,
euclidean-planes-subtype,
euclidean-plane-subtype,
euclidean-plane-structure-subtype,
geo-point_wf,
geo-intersect_wf,
geoline-subtype1,
geo-incident_wf,
geo-intersect-lines-iff,
geo-incident-line,
geo-plsep_wf,
basic-geometry_wf,
euclidean-plane-subtype-basic,
geo-intersection-unicity,
geo-sep_wf,
pi1_wf_top,
geo-sep-or,
geo-sep-sym,
geo-colinear_wf,
lelt_wf,
false_wf,
length_of_nil_lemma,
length_of_cons_lemma,
geo-colinear-is-colinear-set,
not-lsep-iff-colinear,
geo-line-pt-sep,
euclidean-plane-axioms,
geo-colinear-transitivity
Rules used in proof :
dependent_functionElimination,
independent_isectElimination,
instantiate,
because_Cache,
sqequalRule,
applyEquality,
hypothesis,
hypothesisEquality,
rename,
setElimination,
isectElimination,
extract_by_obid,
introduction,
cut,
productEquality,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
universeIsType,
unionElimination,
dependent_set_memberEquality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality
Latex:
\mforall{}eu:EuclideanParPlane. \mforall{}l,m:Line. \mforall{}x,y:Point.
(l \mbackslash{}/ m {}\mRightarrow{} (x I l \mwedge{} x I m) {}\mRightarrow{} (y I l \mwedge{} y I m) {}\mRightarrow{} x \mequiv{} y)
Date html generated:
2019_10_16-PM-02_42_35
Last ObjectModification:
2019_08_30-AM-07_09_19
Theory : euclidean!plane!geometry
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