Nuprl Lemma : derivative-function-rmul-const
∀f,f':ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on (-∞, ∞)
⇒ (∀a:ℝ. d(f[a * x])/dx = λx.a * f'[a * x] on (-∞, ∞)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
req: x = y,
rmul: a * b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
uimplies: b supposing a,
label: ...$L... t,
uiff: uiff(P;Q),
and: P ∧ Q,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
simple-chain-rule,
riiint_wf,
rmul_wf,
real_wf,
i-member_wf,
top_wf,
member_riiint_lemma,
subtype_rel_dep_function,
true_wf,
subtype_rel_self,
set_wf,
iproper-riiint,
req_weakening,
req_wf,
derivative_wf,
all_wf,
int-to-real_wf,
derivative-mul,
derivative-const,
derivative-id,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
derivative_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesisEquality,
setElimination,
rename,
setEquality,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
functionExtensionality,
functionEquality,
natural_numberEquality,
dependent_set_memberEquality,
productElimination,
approximateComputation,
int_eqEquality,
intEquality
Latex:
\mforall{}f,f':\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} (\mforall{}a:\mBbbR{}. d(f[a * x])/dx = \mlambda{}x.a * f'[a * x] on (-\minfty{}, \minfty{})))
Date html generated:
2017_10_03-PM-00_19_29
Last ObjectModification:
2017_08_26-PM-01_39_04
Theory : reals
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