Nuprl Lemma : geo-gt-implies-point
∀e:EuclideanPlane. ∀a,b,c,d:Point. (ab > cd ⇒ c ≠ d ⇒ (¬¬(∃f:Point. (c-d-f ∧ cf ≅ ab))))
Proof
Definitions occuring in Statement :
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-gt: cd > ab,
geo-congruent: ab ≅ cd,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
geo-gt: cd > ab,
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
uimplies: b supposing a,
basic-geometry-: BasicGeometry-,
uiff: uiff(P;Q),
cand: A c∧ B,
prop: ℙ,
true: True
Lemmas referenced :
geo-proper-extend-exists,
geo-O_wf,
geo-X_wf,
geo-sep-sym,
geo-sep-O-X,
geo-strict-between-sep3,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-congruent-sep,
geo-construction-unicity,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-congruent-iff-length,
geo-eq_inversion,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-strict-between_wf,
geo-congruent_wf,
istype-void,
geo-sep_wf,
geo-gt_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalHypSubstitution,
imageElimination,
productElimination,
introduction,
extract_by_obid,
dependent_functionElimination,
sqequalRule,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_functionElimination,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
equalityTransitivity,
dependent_pairFormation_alt,
independent_pairFormation,
lambdaEquality_alt,
equalitySymmetry,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
voidElimination,
functionIsType
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab > cd {}\mRightarrow{} c \mneq{} d {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}f:Point. (c-d-f \mwedge{} cf \mcong{} ab))))
Date html generated:
2019_10_16-PM-01_17_14
Last ObjectModification:
2019_08_07-PM-02_43_56
Theory : euclidean!plane!geometry
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