Nuprl Lemma : fun-converges-to_wf

[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀[g:I ⟶ℝ].  (lim n→∞.f[n;x] = λy.g[y] for x ∈ I ∈ ℙ)


Proof




Definitions occuring in Statement :  fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I rfun: I ⟶ℝ interval: Interval nat: uall: [x:A]. B[x] prop: so_apply: x[s1;s2] so_apply: x[s] member: t ∈ T function: x:A ⟶ B[x]
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T implies:  Q uall: [x:A]. B[x] prop: fun-converges-to: lim n→∞.f[n; x] = λy.g[y] for x ∈ I so_lambda: λ2x.t[x] nat_plus: + so_apply: x[s1;s2] subtype_rel: A ⊆B rfun: I ⟶ℝ so_apply: x[s] uimplies: supposing a rneq: x ≠ y guard: {T} or: P ∨ Q iff: ⇐⇒ Q and: P ∧ Q rev_implies:  Q int_upper: {i...} decidable: Dec(P) satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False not: ¬A top: Top
Lemmas referenced :  interval_wf nat_wf rfun_wf rless_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_plus_properties int_upper_properties rless-int int-to-real_wf rdiv_wf nat_plus_subtype_nat int_upper_subtype_nat rsub_wf rabs_wf rleq_wf int_upper_wf real_wf exists_wf i-approx_wf icompact_wf nat_plus_wf all_wf i-member_wf i-member-approx
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity dependent_functionElimination thin hypothesisEquality independent_functionElimination hypothesis dependent_set_memberEquality because_Cache isectElimination isect_memberFormation introduction sqequalRule setEquality lambdaEquality lambdaFormation setElimination rename applyEquality natural_numberEquality independent_isectElimination inrFormation productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll axiomEquality equalityTransitivity equalitySymmetry functionEquality

Latex:
\mforall{}[I:Interval].  \mforall{}[f:\mBbbN{}  {}\mrightarrow{}  I  {}\mrightarrow{}\mBbbR{}].  \mforall{}[g:I  {}\mrightarrow{}\mBbbR{}].    (lim  n\mrightarrow{}\minfty{}.f[n;x]  =  \mlambda{}y.g[y]  for  x  \mmember{}  I  \mmember{}  \mBbbP{})



Date html generated: 2016_05_18-AM-09_52_33
Last ObjectModification: 2016_01_17-AM-02_53_18

Theory : reals


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