### Nuprl Lemma : derivative_wf

`∀[I:Interval]. ∀[f,g:I ⟶ℝ].  (λx.g[x] = d(f[x])/dx on I ∈ ℙ)`

Proof

Definitions occuring in Statement :  derivative: `λz.g[z] = d(f[x])/dx on I` rfun: `I ⟶ℝ` interval: `Interval` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` member: `t ∈ T`
Definitions unfolded in proof :  prop: `ℙ` uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` all: `∀x:A. B[x]` derivative: `λz.g[z] = d(f[x])/dx on I` so_lambda: `λ2x.t[x]` and: `P ∧ Q` so_apply: `x[s]` rfun: `I ⟶ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` rneq: `x ≠ y` guard: `{T}` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rless: `x < y` sq_exists: `∃x:{A| B[x]}` 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 rfun_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 rmul_wf rsub_wf rabs_wf rleq_wf int-to-real_wf rless_wf real_wf sq_exists_wf i-approx_wf icompact_wf nat_plus_wf all_wf i-member-approx i-member_wf
Rules used in proof :  isectElimination because_Cache dependent_set_memberEquality hypothesis independent_functionElimination hypothesisEquality thin dependent_functionElimination sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution sqequalHypSubstitution lemma_by_obid cut isect_memberFormation introduction sqequalRule lambdaEquality setEquality lambdaFormation setElimination rename productEquality natural_numberEquality functionEquality applyEquality independent_isectElimination inrFormation productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll axiomEquality equalityTransitivity equalitySymmetry

Latex:
\mforall{}[I:Interval].  \mforall{}[f,g:I  {}\mrightarrow{}\mBbbR{}].    (\mlambda{}x.g[x]  =  d(f[x])/dx  on  I  \mmember{}  \mBbbP{})

Date html generated: 2016_05_18-AM-09_59_10
Last ObjectModification: 2016_01_17-AM-00_41_24

Theory : reals

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