### Nuprl Lemma : derivative-id

`∀[I:Interval]. λx.r1 = d(x)/dx on I`

Proof

Definitions occuring in Statement :  derivative: `λz.g[z] = d(f[x])/dx on I` interval: `Interval` int-to-real: `r(n)` uall: `∀[x:A]. B[x]` natural_number: `\$n`
Definitions unfolded in proof :  so_apply: `x[s]` top: `Top` not: `¬A` false: `False` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` decidable: `Dec(P)` rless: `x < y` or: `P ∨ Q` guard: `{T}` rneq: `x ≠ y` uimplies: `b supposing a` nat_plus: `ℕ+` so_lambda: `λ2x.t[x]` prop: `ℙ` true: `True` less_than': `less_than'(a;b)` squash: `↓T` less_than: `a < b` implies: `P `` Q` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` cand: `A c∧ B` and: `P ∧ Q` member: `t ∈ T` sq_exists: `∃x:{A| B[x]}` all: `∀x:A. B[x]` derivative: `λz.g[z] = d(f[x])/dx on I` uall: `∀[x:A]. B[x]` uiff: `uiff(P;Q)` nat: `ℕ` subtype_rel: `A ⊆r B` rsub: `x - y` absval: `|i|` rev_uimplies: `rev_uimplies(P;Q)` sq_stable: `SqStable(P)` real: `ℝ`
Lemmas referenced :  rleq_functionality rmul-nonneg-case1 rleq-int-fractions2 sq_stable__less_than sq_stable__icompact decidable__le intformle_wf itermMultiply_wf int_formula_prop_le_lemma int_term_value_mul_lemma zero-rleq-rabs rabs-abs radd-zero-both rminus-zero rmul-zero-both radd-int rmul_functionality rminus-as-rmul rmul-distrib2 rmul-identity1 radd-ac radd-assoc req_inversion rminus-radd rmul-one-both req_weakening rmul_over_rminus rmul-distrib req_transitivity rminus_functionality radd_functionality rabs_functionality req_functionality uiff_transitivity req_wf radd_wf rminus_wf absval_wf nat_wf req-int decidable__equal_int intformeq_wf int_formula_prop_eq_lemma int-to-real_wf rless-int rleq_wf rabs_wf rsub_wf i-member_wf i-approx_wf real_wf rless_wf all_wf less_than_wf rmul_wf rdiv_wf 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 set_wf nat_plus_wf icompact_wf interval_wf
Rules used in proof :  computeAll voidEquality voidElimination isect_memberEquality intEquality int_eqEquality dependent_pairFormation unionElimination inrFormation independent_isectElimination dependent_set_memberEquality functionEquality because_Cache lambdaEquality productEquality rename setElimination baseClosed hypothesisEquality imageMemberEquality introduction independent_pairFormation sqequalRule independent_functionElimination productElimination dependent_functionElimination hypothesis natural_numberEquality thin isectElimination sqequalHypSubstitution lemma_by_obid cut dependent_set_memberFormation lambdaFormation isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution applyEquality addEquality minusEquality multiplyEquality imageElimination

Latex:
\mforall{}[I:Interval].  \mlambda{}x.r1  =  d(x)/dx  on  I

Date html generated: 2016_05_18-AM-10_06_01
Last ObjectModification: 2016_01_17-AM-00_39_15

Theory : reals

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