Nuprl Lemma : dist-fun_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[x:X]. (dist-fun(d;x) ∈ FUN(X ⟶ ℝ))
Proof
Definitions occuring in Statement :
dist-fun: dist-fun(d;x),
mfun: FUN(X ⟶ Y),
rmetric: rmetric(),
metric: metric(X),
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
is-mfun: f:FUN(X;Y),
dist-fun: dist-fun(d;x),
mfun: FUN(X ⟶ Y),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
meq_weakening,
mdist_functionality,
req_functionality,
req_weakening,
rmetric-meq,
istype-universe,
metric_wf,
rmetric_wf,
real_wf,
is-mfun_wf,
meq_wf,
mdist_wf
Rules used in proof :
independent_isectElimination,
productElimination,
universeEquality,
instantiate,
isectIsTypeImplies,
isect_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
because_Cache,
universeIsType,
lambdaFormation_alt,
inhabitedIsType,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaEquality_alt,
sqequalRule,
dependent_set_memberEquality_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x:X]. (dist-fun(d;x) \mmember{} FUN(X {}\mrightarrow{} \mBbbR{}))
Date html generated:
2019_10_30-AM-07_11_40
Last ObjectModification:
2019_10_25-PM-04_32_59
Theory : reals
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