Nuprl Lemma : rat-cube-complex-polyhedron-metric-subspace
∀[k:ℕ]. ∀[K:ℚCube(k) List]. metric-subspace(ℝ^k;rn-prod-metric(k);|K|)
Proof
Definitions occuring in Statement :
rat-cube-complex-polyhedron: |K|,
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
metric-subspace: metric-subspace(X;d;A),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
rational-cube: ℚCube(k)
Definitions unfolded in proof :
metric-subspace: metric-subspace(X;d;A),
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
rev_implies: P ⇐ Q,
false: False,
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
rat-cube-complex-polyhedron: |K|,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
list_wf,
rat-cube-complex-polyhedron_wf,
strong-subtype_witness,
meq_wf,
istype-void,
in-rat-cube_functionality,
iff_weakening_uiff,
l_exists_functionality,
l_member_wf,
in-rat-cube_wf,
rational-cube_wf,
l_exists_wf,
not_wf,
rn-prod-metric_wf,
real-vec_wf,
set-metric-subspace
Rules used in proof :
isectIsTypeImplies,
isect_memberEquality_alt,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairEquality,
inhabitedIsType,
functionIsType,
voidElimination,
productElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation_alt,
independent_isectElimination,
universeIsType,
setIsType,
rename,
setElimination,
lambdaEquality_alt,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:\mBbbQ{}Cube(k) List]. metric-subspace(\mBbbR{}\^{}k;rn-prod-metric(k);|K|)
Date html generated:
2019_10_31-AM-06_04_04
Last ObjectModification:
2019_10_30-PM-04_25_58
Theory : real!vectors
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