Nuprl Lemma : rccp-dist-0
∀[k,n:ℕ]. ∀[K:{K:n-dim-complex| 0 < ||K||} ]. ∀[x:ℝ^k]. uiff(dist(x, |K|) = r0;x ∈ |K|)
Proof
Definitions occuring in Statement :
rccp-dist: dist(x, |K|),
rat-cube-complex-polyhedron: |K|,
real-vec: ℝ^n,
req: x = y,
int-to-real: r(n),
length: ||as||,
nat: ℕ,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
respects-equality: respects-equality(S;T),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
rat-cube-complex-polyhedron: |K|,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
rccp-dist: dist(x, |K|),
rational-cube-complex: n-dim-complex,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
length_wf,
istype-less_than,
rational-cube-complex_wf,
real-vec_wf,
req_witness,
int-to-real_wf,
rccp-dist_wf,
req_wf,
l_member_wf,
in-rat-cube_wf,
rational-cube_wf,
l_exists_wf,
not_wf,
respects-equality-set-trivial,
rccp-compact_wf,
rat-cube-complex-polyhedron-metric-subspace,
rat-cube-complex-polyhedron_wf,
mcomplete-rn-prod-metric,
rn-prod-metric_wf,
compact-dist-zero-in-complete
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
promote_hyp,
natural_numberEquality,
independent_isectElimination,
inhabitedIsType,
setIsType,
lambdaEquality_alt,
universeIsType,
equalityIstype,
axiomEquality,
sqequalRule,
equalitySymmetry,
equalityTransitivity,
isect_memberFormation_alt,
independent_pairFormation,
productElimination,
rename,
setElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
thin,
isectElimination,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[K:\{K:n-dim-complex| 0 < ||K||\} ]. \mforall{}[x:\mBbbR{}\^{}k]. uiff(dist(x, |K|) = r0;x \mmember{} |K|)
Date html generated:
2019_10_31-AM-06_04_20
Last ObjectModification:
2019_10_30-PM-04_27_41
Theory : real!vectors
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