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

`∀I:Interval. ∀f:I ⟶ℝ.  (icompact(I) `` f[x] continuous for x ∈ I `` totally-bounded(f[x](x∈I)))`

Proof

Definitions occuring in Statement :  continuous: `f[x] continuous for x ∈ I` rrange: `f[x](x∈I)` icompact: `icompact(I)` rfun: `I ⟶ℝ` interval: `Interval` totally-bounded: `totally-bounded(A)` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` true: `True` and: `P ∧ Q` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` label: `...\$L... t` rfun: `I ⟶ℝ` icompact: `icompact(I)`
Lemmas referenced :  continuous-range-totally-bounded less_than_wf i-nonvoid_wf squash_wf true_wf i-approx-of-compact iff_weakening_equal continuous_wf i-member_wf real_wf icompact_wf rfun_wf interval_wf set_wf subtype_rel-equal i-approx_wf equal_wf totally-bounded_wf rset_wf rrange_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality independent_functionElimination hypothesis dependent_set_memberEquality natural_numberEquality sqequalRule independent_pairFormation imageMemberEquality baseClosed isectElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry because_Cache universeEquality independent_isectElimination productElimination setElimination rename setEquality functionEquality instantiate

Latex:
\mforall{}I:Interval.  \mforall{}f:I  {}\mrightarrow{}\mBbbR{}.    (icompact(I)  {}\mRightarrow{}  f[x]  continuous  for  x  \mmember{}  I  {}\mRightarrow{}  totally-bounded(f[x](x\mmember{}I)))

Date html generated: 2017_10_03-AM-10_23_42
Last ObjectModification: 2017_07_28-AM-08_07_54

Theory : reals

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