Nuprl Lemma : totally-bounded-bounded-above
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ bounded-above(A))
Proof
Definitions occuring in Statement :
totally-bounded: totally-bounded(A),
bounded-above: bounded-above(A),
rset: Set(ℝ),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
bounded-above: bounded-above(A),
totally-bounded: totally-bounded(A),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exists: ∃x:A. B[x],
prop: ℙ,
nat_plus: ℕ+,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s],
upper-bound: A ≤ b,
cand: A c∧ B,
guard: {T},
le: A ≤ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
sq_stable: SqStable(P),
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
int-to-real_wf,
rless-int,
real_wf,
rless_wf,
nat_plus_wf,
int_seg_wf,
rset-member_wf,
rabs_wf,
rsub_wf,
rset_wf,
radd_wf,
rmaximum_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract-add-cancel,
decidable__lt,
istype-le,
istype-less_than,
upper-bound_wf,
rabs-bounds,
rless_transitivity2,
radd-preserves-rless,
itermAdd_wf,
squash_wf,
true_wf,
radd_comm_eq,
subtype_rel_self,
iff_weakening_equal,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
radd-preserves-rleq,
rminus_wf,
sq_stable__less_than,
int_seg_properties,
itermMinus_wf,
rleq_functionality,
real_term_value_minus_lemma,
rmaximum_ub,
subtype_base_sq,
set_subtype_base,
less_than_wf,
int_subtype_base,
add-associates,
add-swap,
add-commutes,
zero-add,
rless_transitivity1,
rleq_weakening_rless
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
functionIsType,
universeIsType,
productIsType,
setElimination,
rename,
because_Cache,
applyEquality,
dependent_pairFormation_alt,
closedConclusion,
independent_isectElimination,
unionElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
addEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
instantiate,
universeEquality,
cumulativity,
intEquality
Latex:
\mforall{}[A:Set(\mBbbR{})]. (totally-bounded(A) {}\mRightarrow{} bounded-above(A))
Date html generated:
2019_10_29-AM-10_43_50
Last ObjectModification:
2019_04_19-PM-06_12_46
Theory : reals
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