Nuprl Lemma : geo-add-length-le-implies-eq
∀[e:BasicGeometry]. ∀[x:Length]. ∀[a,b:Point]. a ≡ b supposing x + |ab| ≤ x
Proof
Definitions occuring in Statement :
geo-add-length: p + q,
geo-le: p ≤ q,
geo-length: |s|,
geo-length-type: Length,
geo-mk-seg: ab,
basic-geometry: BasicGeometry,
geo-eq: a ≡ b,
geo-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
false: False,
not: ¬A,
geo-eq: a ≡ b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
basic-geometry: BasicGeometry,
prop: ℙ,
squash: ↓T,
geo-zero-length: 0,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
all: ∀x:A. B[x]
Lemmas referenced :
geo-length-type_wf,
geo-point_wf,
Error :basic-geo-primitives_wf,
Error :basic-geo-structure_wf,
basic-geometry_wf,
subtype_rel_transitivity,
basic-geometry-subtype,
geo-sep_wf,
iff_weakening_equal,
geo-add-length-zero,
geo-mk-seg_wf,
geo-length_wf,
geo-add-length_wf,
true_wf,
squash_wf,
geo-le_wf,
geo-zero-length_wf,
geo-add-length-cancel-left-le,
geo-le-zero,
geo-zero-length-iff
Rules used in proof :
voidElimination,
isect_memberEquality,
instantiate,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
universeEquality,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
rename,
setElimination,
because_Cache,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[e:BasicGeometry]. \mforall{}[x:Length]. \mforall{}[a,b:Point]. a \mequiv{} b supposing x + |ab| \mleq{} x
Date html generated:
2017_10_02-PM-06_18_33
Last ObjectModification:
2017_08_05-PM-04_13_12
Theory : euclidean!plane!geometry
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