Nuprl Lemma : geo-out-unicity
∀e:BasicGeometry. ∀a,b,c:Point. (out(a bc) ⇒ ab ≅ ac ⇒ b ≡ c)
Proof
Definitions occuring in Statement :
geo-out: out(p ab),
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-congruent: ab ≅ cd,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
false: False,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
not: ¬A,
geo-eq: a ≡ b,
and: P ∧ Q,
geo-out: out(p ab),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
geo-add-length_wf,
geo-eq_inversion,
geo-mk-seg_wf,
geo-length_wf,
geo-point_wf,
geo-out_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-congruent_wf,
geo-sep_wf,
geo-between_wf,
geo-add-length-implies-eq,
iff_weakening_equal,
geo-length-type_wf,
true_wf,
squash_wf,
equal_wf,
geo-congruent-iff-length,
geo-add-length-between
Rules used in proof :
instantiate,
voidElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
universeEquality,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
hypothesis,
independent_isectElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
cut,
independent_pairFormation,
independent_functionElimination,
introduction,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}a,b,c:Point. (out(a bc) {}\mRightarrow{} ab \00D0 ac {}\mRightarrow{} b \mequiv{} c)
Date html generated:
2017_10_02-PM-06_27_34
Last ObjectModification:
2017_08_05-PM-04_40_43
Theory : euclidean!plane!geometry
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