Nuprl Lemma : geo-lt-null-segment
∀e:BasicGeometry. ∀[p:Length]. ∀[a:Point]. uiff(p < |aa|;False)
Proof
Definitions occuring in Statement :
geo-lt: p < q,
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-point: Point,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
false: False
Definitions unfolded in proof :
geo-zero-length: 0,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
basic-geometry: BasicGeometry,
exists: ∃x:A. B[x],
geo-lt: p < q,
false: False,
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
geo-eq: a ≡ b
Lemmas referenced :
geo-length-type_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-point_wf,
false_wf,
geo-lt_wf,
geo-add-length-is-zero,
iff_weakening_equal,
geo-length-null-segment,
true_wf,
squash_wf,
equal_wf,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length_wf,
geo-le-null-segment,
geo-zero-length-iff
Rules used in proof :
instantiate,
voidElimination,
independent_functionElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
independent_isectElimination,
hypothesis,
because_Cache,
rename,
setElimination,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
independent_pairFormation,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}e:BasicGeometry. \mforall{}[p:Length]. \mforall{}[a:Point]. uiff(p < |aa|;False)
Date html generated:
2017_10_02-PM-06_19_19
Last ObjectModification:
2017_08_05-PM-04_13_48
Theory : euclidean!plane!geometry
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