Nuprl Lemma : derivative-function-rminus
∀f,f':ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f'[x] = f'[y])))
⇒ d(f[x])/dx = λx.f'[x] on (-∞, ∞)
⇒ d(f[-(x)])/dx = λx.-(f'[-(x)]) on (-∞, ∞))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
riiint: (-∞, ∞),
req: x = y,
rminus: -(x),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
uimplies: b supposing a,
label: ...$L... t,
true: True,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
squash: ↓T,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
simple-chain-rule,
riiint_wf,
rminus_wf,
real_wf,
i-member_wf,
int-to-real_wf,
member_riiint_lemma,
subtype_rel_dep_function,
true_wf,
subtype_rel_self,
set_wf,
iproper-riiint,
req_weakening,
req_wf,
derivative_wf,
all_wf,
derivative-minus,
derivative-id,
rmul_wf,
derivative_functionality,
uiff_transitivity3,
squash_wf,
rminus-int,
uiff_transitivity,
req_functionality,
rmul-minus,
rmul_over_rminus,
rminus_functionality,
rmul-one-both
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
setEquality,
natural_numberEquality,
because_Cache,
applyEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_functionElimination,
functionExtensionality,
functionEquality,
minusEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}f,f':\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f'[x] = f'[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on (-\minfty{}, \minfty{})
{}\mRightarrow{} d(f[-(x)])/dx = \mlambda{}x.-(f'[-(x)]) on (-\minfty{}, \minfty{}))
Date html generated:
2016_10_26-AM-11_31_12
Last ObjectModification:
2016_09_05-AM-10_19_58
Theory : reals
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