Nuprl Lemma : derivative-add-const
∀[I:Interval]. ∀[C:ℝ]. ∀[f,g:I ⟶ℝ]. (d(f[x])/dx = λx.g[x] on I ⇒ d(C + f[x])/dx = λx.g[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
radd: a + b,
real: ℝ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
all: ∀x:A. B[x],
and: P ∧ Q,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative_wf,
i-member_wf,
real_wf,
rfun_wf,
interval_wf,
top_wf,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
int-to-real_wf,
radd_wf,
req_weakening,
radd-zero-both,
derivative-add,
derivative-const,
derivative_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
setEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I:Interval]. \mforall{}[C:\mBbbR{}]. \mforall{}[f,g:I {}\mrightarrow{}\mBbbR{}]. (d(f[x])/dx = \mlambda{}x.g[x] on I {}\mRightarrow{} d(C + f[x])/dx = \mlambda{}x.g[x] on I)
Date html generated:
2018_05_22-PM-02_46_01
Last ObjectModification:
2017_10_23-AM-00_50_15
Theory : reals
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