Nuprl Lemma : derivative-rminus
d(-(x))/dx = λx.r(-1) on (-∞, ∞)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
rminus: -(x),
int-to-real: r(n),
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
derivative-minus,
riiint_wf,
real_wf,
i-member_wf,
int-to-real_wf,
derivative-id,
rminus_wf,
req_weakening,
set_wf,
itermSubtract_wf,
itermMinus_wf,
itermConstant_wf,
req-iff-rsub-is-0,
derivative_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
setEquality,
isectElimination,
natural_numberEquality,
because_Cache,
independent_functionElimination,
minusEquality,
independent_isectElimination,
productElimination,
lambdaFormation,
approximateComputation,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
d(-(x))/dx = \mlambda{}x.r(-1) on (-\minfty{}, \minfty{})
Date html generated:
2017_10_03-PM-00_12_12
Last ObjectModification:
2017_06_24-PM-00_17_17
Theory : reals
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