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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