Nuprl Lemma : m-closed-subspace_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. m-closed-subspace(X;d;A) ∈ ℙ supposing metric-subspace(X;d;A)
Proof
Definitions occuring in Statement :
m-closed-subspace: m-closed-subspace(X;d;A),
metric-subspace: metric-subspace(X;d;A),
metric: metric(X),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
m-closed-subspace: m-closed-subspace(X;d;A),
implies: P ⇒ Q,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
metric-subspace: metric-subspace(X;d;A),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
mdist_wf,
rleq_wf,
nat_plus_wf,
equal-wf,
istype-universe,
metric_wf,
metric-subspace_wf,
strong-subtype-iff-respects-equality
Rules used in proof :
equalityIstype,
independent_pairFormation,
voidElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_functionElimination,
inrFormation_alt,
because_Cache,
rename,
setElimination,
natural_numberEquality,
closedConclusion,
lambdaFormation_alt,
applyEquality,
productEquality,
functionEquality,
independent_functionElimination,
universeEquality,
instantiate,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
independent_isectElimination,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[A:Type]. m-closed-subspace(X;d;A) \mmember{} \mBbbP{} supposing metric-subspace(X;d;A)
Date html generated:
2019_10_30-AM-06_32_07
Last ObjectModification:
2019_10_23-PM-06_23_57
Theory : reals
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