Nuprl Lemma : prod-metric-space-complete
∀k:ℕ. ∀X:ℕk ⟶ MetricSpace. ((∀i:ℕk. mcomplete(X i)) ⇒ mcomplete(prod-metric-space(k;X)))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
prod-metric-space: prod-metric-space(k;X),
metric-space: MetricSpace,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
metric-space: MetricSpace,
pi1: fst(t),
nat: ℕ,
so_apply: x[s],
pi2: snd(t),
prod-metric-space: prod-metric-space(k;X),
mcomplete: mcomplete(M),
subtype_rel: A ⊆r B,
guard: {T},
metric: metric(X),
prop: ℙ,
mcauchy: mcauchy(d;n.x[n]),
sq_exists: ∃x:A [B[x]],
mdist: mdist(d;x;y),
prod-metric: prod-metric(k;d),
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
mconverges: x[n]↓ as n→∞,
mconverges-to: lim n→∞.x[n] = y,
sq_type: SQType(T),
top: Top,
less_than': less_than'(a;b),
subtract: n - m,
true: True,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
l_exists: (∃x∈L. P[x]),
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
int_upper: {i...},
nequal: a ≠ b ∈ T ,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rtermMultiply: left "*" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rat_term_to_real: rat_term_to_real(f;t)
Latex:
\mforall{}k:\mBbbN{}. \mforall{}X:\mBbbN{}k {}\mrightarrow{} MetricSpace. ((\mforall{}i:\mBbbN{}k. mcomplete(X i)) {}\mRightarrow{} mcomplete(prod-metric-space(k;X)))
Date html generated:
2020_05_20-AM-11_58_01
Last ObjectModification:
2020_01_06-PM-00_17_34
Theory : reals
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