Nuprl Lemma : derivative-minus
∀I:Interval. ∀f,g:I ⟶ℝ. (λx.g[x] = d(f[x])/dx on I ⇒ λx.-(g[x]) = d(-(f[x]))/dx on I)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx 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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
uimplies: b supposing a,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
derivative-const-mul,
int-to-real_wf,
derivative_wf,
real_wf,
i-member_wf,
rfun_wf,
interval_wf,
rmul_wf,
rminus_wf,
req_wf,
req_weakening,
set_wf,
derivative_functionality,
uiff_transitivity,
req_functionality,
rminus-as-rmul,
req_inversion
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
lambdaFormation,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setEquality,
because_Cache,
independent_isectElimination,
productElimination
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}. (\mlambda{}x.g[x] = d(f[x])/dx on I {}\mRightarrow{} \mlambda{}x.-(g[x]) = d(-(f[x]))/dx on I)
Date html generated:
2016_05_18-AM-10_07_08
Last ObjectModification:
2015_12_27-PM-11_03_44
Theory : reals
Home
Index