Nuprl Lemma : geo-gt-implies-point2
∀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-gt: cd > ab,
geo-congruent: ab ≅ cd,
geo-between: a_b_c,
geo-sep: a ≠ b,
geo-point: Point,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
member: t ∈ T,
basic-geometry: BasicGeometry,
euclidean-plane: EuclideanPlane,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
prop: ℙ,
cand: A c∧ B,
stable: Stable{P},
uiff: uiff(P;Q),
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
geo-strict-between: a-b-c
Lemmas referenced :
geo-gt-sep,
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-sep_wf,
geo-gt_wf,
geo-point_wf,
geo-gt-implies-point,
stable__geo-between,
double-negation-hyp-elim,
geo-strict-between_wf,
geo-congruent_wf,
not_wf,
geo-between_wf,
geo-construction-unicity,
geo-strict-between-implies-between,
geo-between-symmetry,
geo-between-outer-trans,
geo-congruent-iff-length,
istype-void,
geo-congruent_functionality,
geo-eq_weakening,
geo-strict-between_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
thin,
sqequalHypSubstitution,
sqequalRule,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
setElimination,
rename,
because_Cache,
productElimination,
applyEquality,
instantiate,
isectElimination,
independent_isectElimination,
universeIsType,
inhabitedIsType,
dependent_pairFormation_alt,
productEquality,
equalityTransitivity,
equalitySymmetry,
productIsType,
independent_pairFormation,
functionIsType,
voidElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (ab > cd {}\mRightarrow{} c \mneq{} d {}\mRightarrow{} (\mexists{}f:Point. (c\_d\_f \mwedge{} cf \mcong{} ab)))
Date html generated:
2019_10_16-PM-01_17_22
Last ObjectModification:
2019_08_07-PM-02_53_40
Theory : euclidean!plane!geometry
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