Nuprl Lemma : totally-bounded-neg
∀[A:Set(ℝ)]. (totally-bounded(A) ⇐⇒ totally-bounded(-(A)))
Proof
Definitions occuring in Statement :
rset-neg: -(A),
totally-bounded: totally-bounded(A),
rset: Set(ℝ),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
totally-bounded: totally-bounded(A),
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
top: Top,
rev_implies: P ⇐ Q,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
rminus_wf,
int_seg_wf,
member_rset_neg_lemma,
istype-void,
rset-member_wf,
squash_wf,
true_wf,
real_wf,
rminus-rminus-eq,
subtype_rel_self,
iff_weakening_equal,
rless_wf,
rabs-rminus,
rsub_wf,
rabs_wf,
rset-neg_wf,
int-to-real_wf,
totally-bounded_wf,
rset_wf,
radd_wf,
itermSubtract_wf,
itermMinus_wf,
itermVar_wf,
itermAdd_wf,
rless_functionality,
rabs_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
universeIsType,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
isect_memberEquality_alt,
voidElimination,
promote_hyp,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
inhabitedIsType,
productIsType,
functionIsType,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[A:Set(\mBbbR{})]. (totally-bounded(A) \mLeftarrow{}{}\mRightarrow{} totally-bounded(-(A)))
Date html generated:
2019_10_29-AM-10_44_44
Last ObjectModification:
2019_04_19-PM-06_33_21
Theory : reals
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