Nuprl Lemma : chain-rule_0
∀I,J:Interval. ∀f,f':I ⟶ℝ. ∀g,g':J ⟶ℝ.
(iproper(J)
⇒ maps-compact(I;J;x.f[x])
⇒ f[x] (proper)continuous for x ∈ I
⇒ f'[x] (proper)continuous for x ∈ I
⇒ g'[x] (proper)continuous for x ∈ J
⇒ d(f[x])/dx = λx.f'[x] on I
⇒ d(g[x])/dx = λx.g'[x] on J
⇒ d(g[f[x]])/dx = λx.g'[f[x]] * f'[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
maps-compact: maps-compact(I;J;x.f[x]),
proper-continuous: f[x] (proper)continuous for x ∈ I,
rfun: I ⟶ℝ,
iproper: iproper(I),
interval: Interval,
rmul: a * b,
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,
derivative: d(f[x])/dx = λz.g[z] on I,
maps-compact-proper: maps-compact-proper(I;J;x.f[x]),
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
and: P ∧ Q,
squash: ↓T,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
sq_exists: ∃x:A [B[x]],
proper-continuous: f[x] (proper)continuous for x ∈ I,
cand: A c∧ B,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rless: x < y,
real: ℝ,
req_int_terms: t1 ≡ t2,
rdiv: (x/y)
Lemmas referenced :
proper-maps-compact,
real_wf,
i-member_wf,
derivative_wf,
proper-continuous_wf,
maps-compact_wf,
iproper_wf,
rfun_wf,
interval_wf,
sq_stable__iproper,
i-approx_wf,
proper-continuous-implies,
istype-less_than,
sq_stable__icompact,
icompact_wf,
nat_plus_wf,
Inorm-bound,
rfun_subtype,
i-approx-is-subinterval,
Inorm_wf,
rleq_wf,
rabs_wf,
i-member-approx,
r-bound-property,
mul_nat_plus,
r-bound_wf,
subtype_rel_self,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rleq-int-fractions2,
sq_stable__and,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
small-reciprocal-real,
sq_stable__rless,
le_witness_for_triv,
rsub_wf,
rmul_wf,
rleq_weakening_rless,
radd_wf,
sq_stable__less_than,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_subtype_base,
radd_functionality_wrt_rleq,
rmin_wf,
rmin_strict_ub,
rmin-rleq,
implies_weakening_uimplies,
itermSubtract_wf,
itermAdd_wf,
r-triangle-inequality,
rleq_functionality,
req_weakening,
rabs_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rminus_wf,
itermMinus_wf,
uimplies_transitivity,
radd_functionality,
req_inversion,
rabs-rmul,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
rleq_weakening,
iff_weakening_uiff,
rmul_comm,
squash_wf,
true_wf,
iff_weakening_equal,
zero-rleq-rabs,
set_subtype_base,
less_than_wf,
rmul_functionality_wrt_rleq2,
rmul_preserves_rleq2,
rinv_wf2,
req_transitivity,
rinv-mul-as-rdiv,
radd-non-neg,
rmul-nonneg-case1,
rmul_preserves_rleq,
rneq_functionality,
rmul-int,
rmul_functionality,
rinv_functionality2,
rinv-of-rmul,
rmul-rinv3,
rdiv_functionality,
radd-preserves-rleq,
rleq-int,
rmul-int-rdiv,
rmul-rinv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setIsType,
universeIsType,
hypothesis,
isectElimination,
independent_functionElimination,
promote_hyp,
because_Cache,
inhabitedIsType,
productElimination,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
productIsType,
independent_isectElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
isectIsType,
closedConclusion,
independent_pairFormation,
functionIsType,
functionEquality,
setEquality,
multiplyEquality,
inrFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
functionIsTypeImplies,
addEquality,
equalityIsType4,
dependent_set_memberFormation_alt,
equalityIsType1,
inlFormation_alt,
instantiate,
universeEquality,
intEquality,
baseApply
Latex:
\mforall{}I,J:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}. \mforall{}g,g':J {}\mrightarrow{}\mBbbR{}.
(iproper(J)
{}\mRightarrow{} maps-compact(I;J;x.f[x])
{}\mRightarrow{} f[x] (proper)continuous for x \mmember{} I
{}\mRightarrow{} f'[x] (proper)continuous for x \mmember{} I
{}\mRightarrow{} g'[x] (proper)continuous for x \mmember{} J
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on I
{}\mRightarrow{} d(g[x])/dx = \mlambda{}x.g'[x] on J
{}\mRightarrow{} d(g[f[x]])/dx = \mlambda{}x.g'[f[x]] * f'[x] on I)
Date html generated:
2019_10_30-AM-09_06_22
Last ObjectModification:
2018_11_13-AM-11_06_33
Theory : reals
Home
Index