### Nuprl Lemma : continuous-range-totally-bounded

`∀I:Interval. ∀f:I ⟶ℝ.`
`  (f[x] continuous for x ∈ I `` (∀m:ℕ+. (i-nonvoid(i-approx(I;m)) `` totally-bounded(f[x](x∈i-approx(I;m))))))`

Proof

Definitions occuring in Statement :  continuous: `f[x] continuous for x ∈ I` rrange: `f[x](x∈I)` i-nonvoid: `i-nonvoid(I)` rfun: `I ⟶ℝ` i-approx: `i-approx(I;n)` interval: `Interval` totally-bounded: `totally-bounded(A)` nat_plus: `ℕ+` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` continuous: `f[x] continuous for x ∈ I` member: `t ∈ T` icompact: `icompact(I)` and: `P ∧ Q` cand: `A c∧ B` uall: `∀[x:A]. B[x]` prop: `ℙ` totally-bounded: `totally-bounded(A)` exists: `∃x:A. B[x]` sq_exists: `∃x:{A| B[x]}` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rfun: `I ⟶ℝ` nat_plus: `ℕ+` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rless: `x < y` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` label: `...\$L... t` sq_stable: `SqStable(P)` rleq: `x ≤ y` rnonneg: `rnonneg(x)` le: `A ≤ B` squash: `↓T` subinterval: `I ⊆ J ` int_seg: `{i..j-}` lelt: `i ≤ j < k` real: `ℝ` less_than: `a < b` full-partition: `full-partition(I;p)` partition: `partition(I)` less_than': `less_than'(a;b)` true: `True` uiff: `uiff(P;Q)` l_all: `(∀x∈L.P[x])` rrange: `f[x](x∈I)` rset-member: `x ∈ A` r-ap: `f(x)`
Lemmas referenced :  i-approx-closed i-approx-finite icompact_wf i-approx_wf small-reciprocal-real rless_wf int-to-real_wf i-approx-is-subinterval interval_wf rfun_subtype rfun_wf all_wf i-member_wf rleq_wf rabs_wf rsub_wf rdiv_wf rless-int nat_plus_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf equal_wf subinterval_wf real_wf i-nonvoid_wf nat_plus_wf continuous_wf less_than'_wf squash_wf sq_stable__and sq_stable__rless sq_stable__all sq_stable__rleq partition-exists r-ap_wf select_wf full-partition_wf int_seg_properties length_wf sq_stable__less_than decidable__le intformle_wf int_formula_prop_le_lemma int_seg_wf rset-member_wf rrange_wf exists_wf length_of_cons_lemma append_wf cons_wf right-endpoint_wf add_nat_plus length_wf_nat nil_wf less_than_wf length-append length_of_nil_lemma add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma false_wf full-partition-point-member req_weakening req_wf mesh-property rless_transitivity2 rleq_functionality rabs_functionality rsub_functionality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution dependent_functionElimination thin dependent_set_memberEquality hypothesisEquality cut hypothesis independent_pairFormation introduction extract_by_obid because_Cache isectElimination natural_numberEquality productElimination setElimination rename applyEquality independent_isectElimination sqequalRule productEquality lambdaEquality functionEquality inrFormation independent_functionElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll equalityTransitivity equalitySymmetry setEquality minusEquality independent_pairEquality axiomEquality imageMemberEquality baseClosed imageElimination addEquality functionExtensionality applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion

Latex:
\mforall{}I:Interval.  \mforall{}f:I  {}\mrightarrow{}\mBbbR{}.
(f[x]  continuous  for  x  \mmember{}  I
{}\mRightarrow{}  (\mforall{}m:\mBbbN{}\msupplus{}.  (i-nonvoid(i-approx(I;m))  {}\mRightarrow{}  totally-bounded(f[x](x\mmember{}i-approx(I;m))))))

Date html generated: 2017_10_03-AM-10_23_25
Last ObjectModification: 2017_07_28-AM-08_07_41

Theory : reals

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