Nuprl Lemma : rccp-dist-positive
∀k:ℕ
∀[n:ℕ]
∀K:{K:n-dim-complex| 0 < ||K||} . ∀x:ℝ^k.
(r0 < dist(x, |K|) ⇐⇒ ∃n:ℕ+. ∀p:|K|. ((r1/r(n)) ≤ mdist(rn-prod-metric(k);x;p)))
Proof
Definitions occuring in Statement :
rccp-dist: dist(x, |K|),
rat-cube-complex-polyhedron: |K|,
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
int-to-real: r(n),
length: ||as||,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
rccp-dist: dist(x, |K|),
rat-cube-complex-polyhedron: |K|,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rational-cube-complex: n-dim-complex,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
length_wf,
istype-less_than,
rational-cube-complex_wf,
real-vec_wf,
rccp-compact_wf,
rat-cube-complex-polyhedron_wf,
rn-prod-metric_wf,
compact-dist-positive
Rules used in proof :
inhabitedIsType,
natural_numberEquality,
setIsType,
universeIsType,
lambdaEquality_alt,
independent_isectElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}
\mforall{}[n:\mBbbN{}]
\mforall{}K:\{K:n-dim-complex| 0 < ||K||\} . \mforall{}x:\mBbbR{}\^{}k.
(r0 < dist(x, |K|) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}p:|K|. ((r1/r(n)) \mleq{} mdist(rn-prod-metric(k);x;p)))
Date html generated:
2019_10_31-AM-06_04_22
Last ObjectModification:
2019_10_30-PM-04_32_30
Theory : real!vectors
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