Nuprl Lemma : mcompact-rat-cube
∀k:ℕ. ∀c:ℚCube(k). ((↑Inhabited(c)) ⇒ mcompact({x:ℝ^k| in-rat-cube(k;x;c)} ;rn-prod-metric(k)))
Proof
Definitions occuring in Statement :
in-rat-cube: in-rat-cube(k;p;c),
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
mcompact: mcompact(X;d),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
squash: ↓T,
sq_stable: SqStable(P),
cand: A c∧ B,
in-rat-cube: in-rat-cube(k;p;c),
so_apply: x[s],
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
top: Top,
ext-eq: A ≡ B,
rn-prod-metric: rn-prod-metric(n),
iff: P ⇐⇒ Q,
guard: {T},
inhabited-rat-interval: Inhabited(I),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
sq_stable__rleq,
subtype_rel_weakening,
subtype_rel_transitivity,
rn-prod-metric_wf,
mcompact_functionality,
real-vec_wf,
in-rat-cube_wf,
rleq_wf,
subtype_rel_dep_function,
istype-void,
member_rccint_lemma,
q_le_wf,
iff_weakening_equal,
assert-q_le-eq,
qle_wf,
assert-inhabited-rat-cube,
rleq-rat2real,
mcompact-interval,
istype-nat,
rational-cube_wf,
inhabited-rat-cube_wf,
istype-assert,
metric-on-subtype,
rmetric_wf,
int_seg_wf,
rat2real_wf,
rccint_wf,
i-member_wf,
real_wf,
mcompact-product
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
functionEquality,
functionExtensionality,
functionIsType,
productIsType,
productEquality,
voidElimination,
isect_memberEquality_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
setIsType,
independent_isectElimination,
rename,
setElimination,
natural_numberEquality,
universeIsType,
because_Cache,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
sqequalRule,
productElimination,
inhabitedIsType,
applyEquality,
isectElimination,
hypothesis,
setEquality,
lambdaEquality_alt,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). ((\muparrow{}Inhabited(c)) {}\mRightarrow{} mcompact(\{x:\mBbbR{}\^{}k| in-rat-cube(k;x;c)\} ;rn-prod-metric(k)))
Date html generated:
2019_10_31-AM-06_03_35
Last ObjectModification:
2019_10_30-AM-11_36_25
Theory : real!vectors
Home
Index