Nuprl Lemma : totally-bounded-sup
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃b:ℝ. sup(A) = b))
Proof
Definitions occuring in Statement :
totally-bounded: totally-bounded(A),
sup: sup(A) = b,
rset: Set(ℝ),
real: ℝ,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
prop: ℙ,
totally-bounded: totally-bounded(A),
exists: ∃x:A. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermAdd: left "+" right,
rtermMultiply: left "*" right,
rtermConstant: "const",
rtermDivide: num "/" denom,
rtermSubtract: left "-" right,
pi2: snd(t),
top: Top,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtract: n - m,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
upper-bound: A ≤ b,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
least-upper-bound,
totally-bounded-implies-nonvoid,
totally-bounded-bounded-above,
rless_wf,
real_wf,
totally-bounded_wf,
rset_wf,
rdiv_wf,
rsub_wf,
int-to-real_wf,
rless-int,
rmul_preserves_rless,
assert-rat-term-eq2,
rtermAdd_wf,
rtermVar_wf,
rtermMultiply_wf,
rtermConstant_wf,
rtermDivide_wf,
rtermSubtract_wf,
istype-int,
req_wf,
radd_wf,
rmul_wf,
rminus_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
rinv_wf2,
itermVar_wf,
itermAdd_wf,
minus-one-mul-top,
istype-void,
subtype_base_sq,
int_subtype_base,
nequal_wf,
itermMinus_wf,
rless-implies-rless,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
radd_functionality,
rmul-rinv3,
int-rinv-cancel,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
radd-preserves-rless,
rmaximum-select,
subtract_wf,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
nat_plus_wf,
set_subtype_base,
less_than_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
subtract-add-cancel,
decidable__lt,
istype-le,
istype-less_than,
rmaximum_wf,
int_seg_properties,
int_seg_wf,
rless-cases,
rset-member_wf,
upper-bound_wf,
le_witness_for_triv,
rabs-as-rmax,
rleq-rmax,
rabs_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening_rless,
radd-preserves-rleq,
rleq_functionality,
req_weakening,
rmaximum_ub,
radd_functionality_wrt_rless2,
rless_transitivity2,
rleq_weakening,
req_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
productElimination,
universeIsType,
inhabitedIsType,
dependent_pairFormation_alt,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
lambdaEquality_alt,
int_eqEquality,
approximateComputation,
productIsType,
minusEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
sqequalBase,
setElimination,
rename,
addEquality,
applyEquality,
imageElimination,
unionElimination,
inlFormation_alt,
functionIsTypeImplies
Latex:
\mforall{}[A:Set(\mBbbR{})]. (totally-bounded(A) {}\mRightarrow{} (\mexists{}b:\mBbbR{}. sup(A) = b))
Date html generated:
2019_10_29-AM-10_44_10
Last ObjectModification:
2019_04_19-PM-06_31_24
Theory : reals
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