Nuprl Lemma : geo-lt-or
∀e:EuclideanPlane. ∀a,b,c,d,x,y:Point. (|ab| < |cd| ⇒ (|ab| < |xy| ∨ |xy| < |cd|))
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-mk-seg: ab,
euclidean-plane: EuclideanPlane,
geo-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
euclidean-plane: EuclideanPlane,
uall: ∀[x:A]. B[x],
basic-geometry: BasicGeometry,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
prop: ℙ,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
geo-sep-or,
geo-length_wf1,
geo-mk-seg_wf,
geo-sep-iff-or-lt,
geo-lt_wf,
geo-length_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-point_wf,
geo-lt_transitivity,
geo-le_weakening-lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
isectElimination,
sqequalRule,
hypothesisEquality,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
inlFormation_alt,
universeIsType,
dependent_set_memberEquality_alt,
instantiate,
independent_isectElimination,
unionElimination,
inrFormation_alt
Latex:
\mforall{}e:EuclideanPlane. \mforall{}a,b,c,d,x,y:Point. (|ab| < |cd| {}\mRightarrow{} (|ab| < |xy| \mvee{} |xy| < |cd|))
Date html generated:
2019_10_16-PM-02_33_49
Last ObjectModification:
2019_07_26-AM-11_38_27
Theory : euclidean!plane!geometry
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