Nuprl Lemma : rat-cube-complex-polyhedron-closed
∀[k:ℕ]. ∀[K:ℚCube(k) List]. ∀[v:|K|]. ∀[x:ℝ^k]. x ∈ |K| supposing v ≡ x
Proof
Definitions occuring in Statement :
rat-cube-complex-polyhedron: |K|,
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
meq: x ≡ y,
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
false: False,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
rat-cube-complex-polyhedron: |K|,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
list_wf,
rat-cube-complex-polyhedron_wf,
rn-prod-metric_wf,
real-vec_wf,
meq_wf,
istype-void,
l_exists_wf,
in-rat-cube_functionality,
iff_weakening_uiff,
l_member_wf,
in-rat-cube_wf,
rational-cube_wf,
l_exists_functionality
Rules used in proof :
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
functionIsType,
voidElimination,
productElimination,
independent_isectElimination,
because_Cache,
universeIsType,
setIsType,
lambdaEquality_alt,
sqequalRule,
dependent_functionElimination,
hypothesis,
isectElimination,
extract_by_obid,
independent_functionElimination,
lambdaFormation_alt,
hypothesisEquality,
dependent_set_memberEquality_alt,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:\mBbbQ{}Cube(k) List]. \mforall{}[v:|K|]. \mforall{}[x:\mBbbR{}\^{}k]. x \mmember{} |K| supposing v \mequiv{} x
Date html generated:
2019_10_30-AM-10_13_08
Last ObjectModification:
2019_10_29-PM-01_34_06
Theory : real!vectors
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