Nuprl Lemma : derivative-minus-minus
∀I:Interval. ∀f,g:I ⟶ℝ. (d(f[x])/dx = λx.-(g[x]) on I ⇒ d(-(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,
rminus: -(x),
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uall: ∀[x:A]. B[x],
prop: ℙ,
label: ...$L... t,
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x)
Lemmas referenced :
derivative-minus,
i-member_wf,
real_wf,
rminus_wf,
derivative_wf,
rfun_wf,
interval_wf,
req_weakening,
set_wf,
rminus-rminus,
derivative_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
isectElimination,
setEquality,
because_Cache,
independent_functionElimination,
independent_isectElimination
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}. (d(f[x])/dx = \mlambda{}x.-(g[x]) on I {}\mRightarrow{} d(-(f[x]))/dx = \mlambda{}x.g[x] on I)
Date html generated:
2016_10_26-AM-11_22_33
Last ObjectModification:
2016_08_28-PM-06_56_46
Theory : reals
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