Nuprl Lemma : scale-metric-complete
∀[X:Type]. ∀[d:metric(X)]. ∀c:{c:ℝ| r0 < c} . (mcomplete(X with d) ⇐⇒ mcomplete(X with c*d))
Proof
Definitions occuring in Statement :
mcomplete: mcomplete(M),
mk-metric-space: X with d,
scale-metric: c*d,
metric: metric(X),
rless: x < y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
mcomplete: mcomplete(M),
mk-metric-space: X with d,
rev_implies: P ⇐ Q,
mconverges: x[n]↓ as n→∞,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
metric: metric(X),
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
scale-metric-cauchy,
scale-metric-converges,
mconverges-to_wf,
scale-metric_wf,
istype-nat,
mcauchy_wf,
mcomplete_wf,
mk-metric-space_wf,
subtype_rel_sets_simple,
real_wf,
rless_wf,
int-to-real_wf,
rleq_wf,
rleq_weakening_rless,
metric_wf,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
independent_pairFormation,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
universeIsType,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
functionIsType,
natural_numberEquality,
independent_isectElimination,
setIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}c:\{c:\mBbbR{}| r0 < c\} . (mcomplete(X with d) \mLeftarrow{}{}\mRightarrow{} mcomplete(X with c*d))
Date html generated:
2019_10_30-AM-06_45_37
Last ObjectModification:
2019_10_02-AM-10_57_26
Theory : reals
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