Nuprl Lemma : compact-metric-to-int-bounded
∀[X:Type]. ∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℤ. ∃B:ℕ. ∀x:X. ∃y:X. (x ≡ y ∧ (|f y| ≤ B))
Proof
Definitions occuring in Statement :
m-TB: m-TB(X;d),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
meq: x ≡ y,
metric: metric(X),
absval: |i|,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
m-TB: m-TB(X;d),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
mtb-cantor: mtb-cantor(mtb),
pi1: fst(t),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
exists: ∃x:A. B[x],
cand: A c∧ B,
prop: ℙ,
guard: {T}
Lemmas referenced :
mtb-cantor-map-onto,
general-cantor-to-int-bounded,
pi1_wf_top,
nat_wf,
nat_plus_wf,
subtype_rel_product,
int_seg_wf,
istype-nat,
top_wf,
istype-void,
mtb-cantor-map_wf,
mtb-point-cantor_wf,
meq_inversion,
meq_wf,
istype-le,
absval_wf,
istype-int,
m-TB_wf,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
functionEquality,
hypothesis,
setElimination,
rename,
applyEquality,
sqequalRule,
lambdaEquality_alt,
productEquality,
natural_numberEquality,
universeIsType,
because_Cache,
functionIsType,
independent_isectElimination,
isect_memberEquality_alt,
voidElimination,
productIsType,
closedConclusion,
productElimination,
dependent_pairFormation_alt,
independent_pairFormation,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}cmplt:mcomplete(X with d). \mforall{}mtb:m-TB(X;d). \mforall{}f:X {}\mrightarrow{} \mBbbZ{}.
\mexists{}B:\mBbbN{}. \mforall{}x:X. \mexists{}y:X. (x \mequiv{} y \mwedge{} (|f y| \mleq{} B))
Date html generated:
2019_10_30-AM-07_10_58
Last ObjectModification:
2019_10_09-AM-11_38_12
Theory : reals
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