Nuprl Lemma : mcompact_functionality
∀[X,Y:Type]. ∀d:metric(X). (mcompact(X;d) ⇐⇒ mcompact(Y;d)) supposing X ≡ Y
Proof
Definitions occuring in Statement :
mcompact: mcompact(X;d),
metric: metric(X),
ext-eq: A ≡ B,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
or: P ∨ Q,
rneq: x ≠ y,
nat: ℕ,
spreadn: spread3,
nat_plus: ℕ+,
istype: istype(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
metric: metric(X),
exists: ∃x:A. B[x],
mconverges: x[n]↓ as n→∞,
guard: {T},
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
m-TB: m-TB(X;d),
mk-metric-space: X with d,
mcomplete: mcomplete(M),
mcompact: mcompact(X;d),
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
ext-eq: A ≡ B,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rless_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_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,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
mdist_wf,
rleq_wf,
subtype_rel_self,
nat_plus_wf,
int_seg_wf,
mcauchy_wf,
istype-nat,
real_wf,
subtype_rel_dep_function,
mconverges-to_wf,
nat_wf,
subtype_rel_weakening,
ext-eq_inversion,
metric-on-subtype,
istype-universe,
ext-eq_wf,
metric_wf,
mcompact_wf
Rules used in proof :
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
inrFormation_alt,
addEquality,
closedConclusion,
productEquality,
natural_numberEquality,
dependent_pairEquality_alt,
dependent_set_memberEquality_alt,
functionIsType,
functionEquality,
lambdaEquality_alt,
because_Cache,
setElimination,
dependent_pairFormation_alt,
functionExtensionality,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
universeEquality,
instantiate,
inhabitedIsType,
hypothesisEquality,
isectElimination,
extract_by_obid,
universeIsType,
promote_hyp,
independent_pairFormation,
lambdaFormation_alt,
rename,
hypothesis,
axiomEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}[X,Y:Type]. \mforall{}d:metric(X). (mcompact(X;d) \mLeftarrow{}{}\mRightarrow{} mcompact(Y;d)) supposing X \mequiv{} Y
Date html generated:
2019_10_30-AM-11_21_52
Last ObjectModification:
2019_10_30-AM-10_51_48
Theory : reals
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