Nuprl Lemma : derivative-rdiv-const-alt
∀a:ℝ. (a ≠ r0 ⇒ (∀I:Interval. ∀f,f':I ⟶ℝ. (d(f[x])/dx = λx.a * f'[x] on I ⇒ d((f[x]/a))/dx = λx.f'[x] on I)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
rdiv: (x/y),
rneq: x ≠ y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
false: False,
not: ¬A,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
true: True,
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
and: P ∧ Q,
pi2: snd(t),
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative-rdiv-const,
derivative_wf,
i-member_wf,
rmul_wf,
rfun_wf,
interval_wf,
rneq_wf,
int-to-real_wf,
real_wf,
rdiv_wf,
req_weakening,
assert-rat-term-eq2,
rtermDivide_wf,
rtermMultiply_wf,
rtermVar_wf,
istype-int,
derivative_functionality
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
universeIsType,
isectElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
inhabitedIsType,
because_Cache,
natural_numberEquality,
independent_isectElimination,
int_eqEquality,
approximateComputation,
independent_pairFormation
Latex:
\mforall{}a:\mBbbR{}
(a \mneq{} r0
{}\mRightarrow{} (\mforall{}I:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}.
(d(f[x])/dx = \mlambda{}x.a * f'[x] on I {}\mRightarrow{} d((f[x]/a))/dx = \mlambda{}x.f'[x] on I)))
Date html generated:
2019_10_30-AM-09_05_33
Last ObjectModification:
2019_04_02-AM-09_45_32
Theory : reals
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