### Nuprl Lemma : continuous_functionality_wrt_rfun-eq

`∀I:Interval. ∀[f1,f2:I ⟶ℝ].  (rfun-eq(I;λx.f1[x];λx.f2[x]) `` f1[x] continuous for x ∈ I `` f2[x] continuous for x ∈ I)`

Proof

Definitions occuring in Statement :  continuous: `f[x] continuous for x ∈ I` rfun-eq: `rfun-eq(I;f;g)` rfun: `I ⟶ℝ` interval: `Interval` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` lambda: `λx.A[x]`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` continuous: `f[x] continuous for x ∈ I` member: `t ∈ T` sq_exists: `∃x:{A| B[x]}` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` so_apply: `x[s]` rfun: `I ⟶ℝ` uimplies: `b supposing a` 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)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` label: `...\$L... t` rfun-eq: `rfun-eq(I;f;g)` r-ap: `f(x)` uiff: `uiff(P;Q)`
Lemmas referenced :  req_weakening rsub_functionality rabs_functionality rleq_functionality interval_wf rfun_wf rfun-eq_wf continuous_wf icompact_wf nat_plus_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 rless-int rdiv_wf i-member-approx less_than_wf real_wf all_wf int-to-real_wf rless_wf i-approx_wf i-member_wf rsub_wf rabs_wf rleq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation sqequalHypSubstitution cut hypothesis dependent_functionElimination thin hypothesisEquality setElimination rename introduction dependent_set_memberEquality independent_pairFormation productElimination promote_hyp independent_functionElimination lemma_by_obid isectElimination because_Cache productEquality natural_numberEquality sqequalRule lambdaEquality functionEquality applyEquality independent_isectElimination inrFormation unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll setEquality

Latex:
\mforall{}I:Interval
\mforall{}[f1,f2:I  {}\mrightarrow{}\mBbbR{}].
(rfun-eq(I;\mlambda{}x.f1[x];\mlambda{}x.f2[x])  {}\mRightarrow{}  f1[x]  continuous  for  x  \mmember{}  I  {}\mRightarrow{}  f2[x]  continuous  for  x  \mmember{}  I)

Date html generated: 2016_05_18-AM-09_09_32
Last ObjectModification: 2016_01_17-AM-02_37_03

Theory : reals

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