Nuprl Lemma : geo-perp-unicity
∀e:BasicGeometry. ∀a,b,c,u,v:Point. (a ≠ b ⇒ Colinear(a;b;u) ⇒ Colinear(a;b;v) ⇒ ab ⊥ cu ⇒ ab ⊥ cv ⇒ u ≡ v)
Proof
Definitions occuring in Statement :
geo-perp: ab ⊥ cd,
basic-geometry: BasicGeometry,
geo-colinear: Colinear(a;b;c),
geo-eq: a ≡ b,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
stable: Stable{P},
not: ¬A,
or: P ∨ Q,
false: False,
geo-perp: ab ⊥ cd,
exists: ∃x:A. B[x],
geo-perp-in: ab ⊥x cd,
and: P ∧ Q,
cand: A c∧ B,
basic-geometry: BasicGeometry,
geo-colinear-set: geo-colinear-set(e; L),
l_all: (∀x∈L.P[x]),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
select: L[n],
cons: [a / b],
subtract: n - m,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
euclidean-plane: EuclideanPlane,
basic-geometry-: BasicGeometry-,
geo-eq: a ≡ b
Lemmas referenced :
geo-perp_wf,
geo-colinear_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
basic-geometry-subtype,
subtype_rel_transitivity,
basic-geometry_wf,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-sep_wf,
geo-point_wf,
stable_geo-eq,
false_wf,
or_wf,
not_wf,
geo-eq_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
geo-colinear-same,
right-angle-legs-same,
geo-colinear-is-colinear-set,
length_of_cons_lemma,
length_of_nil_lemma,
lelt_wf,
right-angle_functionality,
geo-eq_weakening,
geo-eq_inversion,
geo-perp-trivial-when-colinear,
geo-sep-sym,
geo-eq_transitivity,
subtype_rel_self,
basic-geometry-_wf,
stable__not,
right-angle-symmetry,
geo-colinear_functionality,
geo-sep_functionality,
geo-perp-in-unicity2,
right-angles-not-complementary
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
because_Cache,
functionEquality,
independent_functionElimination,
unionElimination,
voidElimination,
productElimination,
dependent_functionElimination,
independent_pairFormation,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
promote_hyp,
addLevel,
impliesFunctionality,
levelHypothesis,
impliesLevelFunctionality
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c,u,v:Point.
(a \mneq{} b {}\mRightarrow{} Colinear(a;b;u) {}\mRightarrow{} Colinear(a;b;v) {}\mRightarrow{} ab \mbot{} cu {}\mRightarrow{} ab \mbot{} cv {}\mRightarrow{} u \mequiv{} v)
Date html generated:
2018_05_22-PM-00_05_54
Last ObjectModification:
2018_04_19-AM-01_14_59
Theory : euclidean!plane!geometry
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