Nuprl Lemma : rccp-dist-nonneg
∀[k,n:ℕ]. ∀[K:{K:n-dim-complex| 0 < ||K||} ]. ∀[x:ℝ^k]. (r0 ≤ dist(x, |K|))
Proof
Definitions occuring in Statement :
rccp-dist: dist(x, |K|),
real-vec: ℝ^n,
rleq: x ≤ y,
int-to-real: r(n),
length: ||as||,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
rational-cube-complex: n-dim-complex,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
rccp-dist: dist(x, |K|),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
rational-cube_wf,
length_wf,
istype-less_than,
rational-cube-complex_wf,
le_witness_for_triv,
rccp-compact_wf,
rn-prod-metric_wf,
real-vec_wf,
compact-dist-nonneg
Rules used in proof :
rename,
setElimination,
natural_numberEquality,
setIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
productElimination,
dependent_functionElimination,
lambdaEquality_alt,
sqequalRule,
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[K:\{K:n-dim-complex| 0 < ||K||\} ]. \mforall{}[x:\mBbbR{}\^{}k]. (r0 \mleq{} dist(x, |K|))
Date html generated:
2019_10_31-AM-06_04_17
Last ObjectModification:
2019_10_30-PM-04_34_41
Theory : real!vectors
Home
Index