Nuprl Lemma : m-cont-real-fun_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[f:X ⟶ ℝ]. (m-cont-real-fun(X;d;x.f[x]) ∈ ℙ)
Proof
Definitions occuring in Statement :
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
metric: metric(X),
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
m-cont-real-fun: m-cont-real-fun(X;d;x.f[x]),
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s]
Lemmas referenced :
real_wf,
rless_wf,
int-to-real_wf,
rleq_wf,
mdist_wf,
rabs_wf,
rsub_wf,
metric_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
functionEquality,
setEquality,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
productEquality,
setElimination,
rename,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[f:X {}\mrightarrow{} \mBbbR{}]. (m-cont-real-fun(X;d;x.f[x]) \mmember{} \mBbbP{})
Date html generated:
2019_10_30-AM-06_27_09
Last ObjectModification:
2019_10_02-AM-10_02_25
Theory : reals
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