Nuprl Lemma : derivative_functionality_wrt_subinterval
∀I,J:Interval. ∀[f,f':I ⟶ℝ]. (J ⊆ I ⇒ d(f[x])/dx = λx.f'[x] on I ⇒ d(f[x])/dx = λx.f'[x] on J)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
subinterval: I ⊆ J ,
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
derivative: d(f[x])/dx = λz.g[z] on I,
member: t ∈ T,
and: P ∧ Q,
prop: ℙ,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
cand: A c∧ B,
iproper: iproper(I),
uimplies: b supposing a,
subinterval: I ⊆ J ,
rbetween: x≤y≤z,
guard: {T},
sq_exists: ∃x:{A| B[x]},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top
Lemmas referenced :
compact-subinterval,
i-approx_wf,
icompact_wf,
subinterval_transitivity,
less_than_wf,
i-approx-is-subinterval,
set_wf,
nat_plus_wf,
iproper_wf,
derivative_wf,
i-member_wf,
real_wf,
subinterval_wf,
rfun_wf,
interval_wf,
i-finite_wf,
i-finite-subinterval,
i-member-finite,
left-endpoint_wf,
i-approx-finite,
icompact-endpoints,
right-endpoint_wf,
rless_transitivity2,
rless_transitivity1,
i-member-approx,
rless_wf,
int-to-real_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
introduction,
extract_by_obid,
setElimination,
rename,
dependent_set_memberEquality,
isectElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
setEquality,
independent_pairFormation,
independent_isectElimination,
promote_hyp,
functionEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}I,J:Interval. \mforall{}[f,f':I {}\mrightarrow{}\mBbbR{}]. (J \msubseteq{} I {}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I {}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on J)
Date html generated:
2016_10_26-AM-11_19_12
Last ObjectModification:
2016_08_22-PM-10_08_15
Theory : reals
Home
Index