Nuprl Lemma : mcompact-product
∀k:ℕ. ∀X:ℕk ⟶ Type. ∀d:i:ℕk ⟶ metric(X i). ((∀i:ℕk. mcompact(X i;d i)) ⇒ mcompact(i:ℕk ⟶ (X i);prod-metric(k;d)))
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
prod-metric: prod-metric(k;d),
metric: metric(X),
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,
universe: Type
Definitions unfolded in proof :
metric-space: MetricSpace,
prop: ℙ,
pi2: snd(t),
pi1: fst(t),
prod-metric-space: prod-metric-space(k;X),
mk-metric-space: X with d,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
mcompact: mcompact(X;d),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
prod-metric_wf,
mcomplete_wf,
mk-metric-space_wf,
prod-metric-space-complete,
istype-nat,
istype-universe,
metric_wf,
mcompact_wf,
int_seg_wf,
m-TB-product
Rules used in proof :
functionEquality,
dependent_pairEquality_alt,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
functionExtensionality_alt,
universeEquality,
instantiate,
functionIsType,
because_Cache,
productElimination,
independent_functionElimination,
hypothesis,
rename,
setElimination,
natural_numberEquality,
universeIsType,
applyEquality,
lambdaEquality_alt,
sqequalRule,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}X:\mBbbN{}k {}\mrightarrow{} Type. \mforall{}d:i:\mBbbN{}k {}\mrightarrow{} metric(X i).
((\mforall{}i:\mBbbN{}k. mcompact(X i;d i)) {}\mRightarrow{} mcompact(i:\mBbbN{}k {}\mrightarrow{} (X i);prod-metric(k;d)))
Date html generated:
2019_10_31-AM-05_59_58
Last ObjectModification:
2019_10_30-AM-11_18_09
Theory : reals
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