Nuprl Lemma : derivative-mul-x
∀I:Interval. ∀f:I ⟶ℝ. ∀g:{h:I ⟶ℝ| ∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((h x) = (h y)))} .
(d(f[x])/dx = λx.g[x] on I ⇒ d(x * f[x])/dx = λx.(x * g[x]) + f[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓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,
top: Top
Lemmas referenced :
derivative_wf,
real_wf,
i-member_wf,
req_wf,
rfun_wf,
interval_wf,
int-to-real_wf,
req_weakening,
sq_stable__req,
derivative-mul,
derivative-id,
rmul_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
derivative_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
hypothesis,
setElimination,
rename,
because_Cache,
inhabitedIsType,
functionIsType,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
productElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}g:\{h:I {}\mrightarrow{}\mBbbR{}| \mforall{}x,y:\{t:\mBbbR{}| t \mmember{} I\} . ((x = y) {}\mRightarrow{} ((h x) = (h y)))\} .
(d(f[x])/dx = \mlambda{}x.g[x] on I {}\mRightarrow{} d(x * f[x])/dx = \mlambda{}x.(x * g[x]) + f[x] on I)
Date html generated:
2019_10_30-AM-09_01_42
Last ObjectModification:
2019_01_03-PM-00_06_08
Theory : reals
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