Nuprl Lemma : derivative-rnexp

`∀n:ℕ+. ∀I:Interval.  d(x^n)/dx = λx.r(n) * x^n - 1 on I`

Proof

Definitions occuring in Statement :  derivative: `d(f[x])/dx = λz.g[z] on I` interval: `Interval` rnexp: `x^k1` rmul: `a * b` int-to-real: `r(n)` nat_plus: `ℕ+` all: `∀x:A. B[x]` subtract: `n - m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` label: `...\$L... t` rfun: `I ⟶ℝ` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` so_apply: `x[s]` subtype_rel: `A ⊆r B` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` itermConstant: `"const"` req_int_terms: `t1 ≡ t2` real_term_value: `real_term_value(f;t)` int_term_ind: int_term_ind itermSubtract: `left (-) right` itermMultiply: `left (*) right` uiff: `uiff(P;Q)` rfun-eq: `rfun-eq(I;f;g)` r-ap: `f(x)` rev_uimplies: `rev_uimplies(P;Q)` itermVar: `vvar` itermAdd: `left (+) right`
Lemmas referenced :  interval_wf nat_plus_properties all_wf derivative_wf rnexp_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf real_wf i-member_wf rmul_wf int-to-real_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma nat_plus_wf primrec-wf-nat-plus nat_plus_subtype_nat rnexp_zero_lemma derivative-id false_wf set_wf real_term_polynomial itermMultiply_wf req-iff-rsub-is-0 derivative_functionality rpower-one req_functionality req_weakening derivative-mul rmul_functionality rnexp_functionality req_wf itermAdd_wf int_term_value_add_lemma radd_wf add-subtract-cancel rnexp-add req_inversion subtract-add-cancel req_transitivity radd_functionality radd-int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid hypothesis thin rename sqequalHypSubstitution isectElimination hypothesisEquality setElimination sqequalRule lambdaEquality dependent_set_memberEquality dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll setEquality because_Cache applyEquality productElimination independent_functionElimination functionEquality addEquality comment

Latex:
\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}I:Interval.    d(x\^{}n)/dx  =  \mlambda{}x.r(n)  *  x\^{}n  -  1  on  I

Date html generated: 2017_10_03-PM-00_13_43
Last ObjectModification: 2017_07_28-AM-08_36_35

Theory : reals

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