Nuprl Lemma : compact-metric-to-real-continuity
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). UC(f:X ⟶ ℝ)
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
m-unif-cont: UC(f:X ⟶ Y),
mfun: FUN(X ⟶ Y),
rmetric: rmetric(),
metric: metric(X),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
mcompact: mcompact(X;d),
mfun: FUN(X ⟶ Y),
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
sq_stable: SqStable(P),
is-mfun: f:FUN(X;Y),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
rmetric: rmetric(),
m-unif-cont: UC(f:X ⟶ Y),
mdist: mdist(d;x;y),
m-TB: m-TB(X;d),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
mtb-cantor: mtb-cantor(mtb),
real: ℝ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
sq_exists: ∃x:A [B[x]],
rless: x < y,
guard: {T},
rneq: x ≠ y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
istype: istype(T),
less_than': less_than'(a;b),
top: Top,
ge: i ≥ j ,
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
nat: ℕ,
pi1: fst(t),
rev_uimplies: rev_uimplies(P;Q),
true: True,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rational-approx: (x within 1/n),
rge: x ≥ y
Latex:
\mforall{}[X:Type]. \mforall{}d:metric(X). \mforall{}c:mcompact(X;d). \mforall{}f:FUN(X {}\mrightarrow{} \mBbbR{}). UC(f:X {}\mrightarrow{} \mBbbR{})
Date html generated:
2020_05_20-AM-11_59_47
Last ObjectModification:
2019_12_14-PM-04_48_37
Theory : reals
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