Nuprl Lemma : prod-metric-space_wf
∀[k:ℕ]. ∀[X:ℕk ⟶ MetricSpace]. (prod-metric-space(k;X) ∈ MetricSpace)
Proof
Definitions occuring in Statement :
prod-metric-space: prod-metric-space(k;X),
metric-space: MetricSpace,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prod-metric-space: prod-metric-space(k;X),
metric-space: MetricSpace,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t)
Lemmas referenced :
int_seg_wf,
prod-metric_wf,
metric_wf,
metric-space_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependent_pairEquality_alt,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
hypothesisEquality,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
universeIsType,
axiomEquality,
functionIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} MetricSpace]. (prod-metric-space(k;X) \mmember{} MetricSpace)
Date html generated:
2019_10_29-AM-11_11_35
Last ObjectModification:
2019_10_02-AM-09_52_10
Theory : reals
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