Nuprl Lemma : geo-lt-implies-point
∀e:EuclideanPlane. ∀a,b,c,d:Point. (|ab| < |cd| ⇒ a ≠ b ⇒ (∃f:Point. (a-b-f ∧ af ≅ cd)))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-strict-between: a-b-c,
geo-congruent: ab ≅ cd,
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,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
basic-geometry: BasicGeometry,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
basic-geometry-: BasicGeometry-,
subtype_rel: A ⊆r B,
guard: {T},
euclidean-plane: EuclideanPlane
Lemmas referenced :
geo-lt-implies-gt-strong-1,
geo-proper-extend-exists,
geo-congruent-iff-length,
geo-add-length-between,
geo-add-length_wf,
squash_wf,
true_wf,
geo-length-type_wf,
basic-geometry_wf,
geo-between-symmetry,
geo-strict-between-implies-between,
geo-strict-between_wf,
geo-congruent_wf,
geo-sep_wf,
euclidean-plane-structure-subtype,
euclidean-plane-subtype,
subtype_rel_transitivity,
euclidean-plane_wf,
euclidean-plane-structure_wf,
geo-primitives_wf,
geo-lt_wf,
geo-length_wf,
geo-mk-seg_wf,
geo-point_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
sqequalRule,
because_Cache,
rename,
dependent_pairFormation_alt,
independent_pairFormation,
isectElimination,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productIsType,
instantiate,
setElimination
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d:Point. (|ab| < |cd| {}\mRightarrow{} a \mneq{} b {}\mRightarrow{} (\mexists{}f:Point. (a-b-f \mwedge{} af \mcong{} cd)))
Date html generated:
2019_10_16-PM-01_37_01
Last ObjectModification:
2019_08_07-PM-03_17_52
Theory : euclidean!plane!geometry
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