Nuprl Lemma : rccp-dist_wf
∀[k,n:ℕ]. ∀[K:{K:n-dim-complex| 0 < ||K||} ]. ∀[x:ℝ^k]. (dist(x, |K|) ∈ ℝ)
Proof
Definitions occuring in Statement :
rccp-dist: dist(x, |K|),
real-vec: ℝ^n,
real: ℝ,
length: ||as||,
nat: ℕ,
less_than: a < b,
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 :
rat-cube-complex-polyhedron: |K|,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rational-cube-complex: n-dim-complex,
rccp-dist: dist(x, |K|),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
rational-cube-complex_wf,
rational-cube_wf,
length_wf,
istype-less_than,
rccp-compact_wf,
rat-cube-complex-polyhedron_wf,
rn-prod-metric_wf,
real-vec_wf,
compact-dist_wf
Rules used in proof :
setIsType,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
natural_numberEquality,
dependent_set_memberEquality_alt,
universeIsType,
lambdaEquality_alt,
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
rename,
thin,
setElimination,
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]. (dist(x, |K|) \mmember{} \mBbbR{})
Date html generated:
2019_10_31-AM-06_04_14
Last ObjectModification:
2019_10_30-PM-04_20_05
Theory : real!vectors
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