Nuprl Lemma : chain-rule_1
∀I,J:Interval. ∀f,f':I ⟶ℝ. ∀g,g':J ⟶ℝ.
(maps-compact(I;J;x.f[x])
⇒ f[x] continuous for x ∈ I
⇒ f'[x] continuous for x ∈ I
⇒ g'[x] continuous for x ∈ J
⇒ λx.f'[x] = d(f[x])/dx on I
⇒ λx.g'[x] = d(g[x])/dx on J
⇒ λx.g'[f[x]] * f'[x] = d(g[f[x]])/dx on I)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
maps-compact: maps-compact(I;J;x.f[x]),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rmul: a * b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
maps-compact: maps-compact(I;J;x.f[x]),
derivative: λz.g[z] = d(f[x])/dx on I,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
rneq: x ≠ y,
guard: {T},
true: True,
less_than': less_than'(a;b),
less_than: a < b,
nat_plus: ℕ+,
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rless: x < y,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
cand: A c∧ B,
continuous: f[x] continuous for x ∈ I,
sq_exists: ∃x:{A| B[x]},
real: ℝ,
rsub: x - y
Lemmas referenced :
req-int-fractions,
decidable__equal_int,
rmul-int-rdiv,
rmul_functionality_wrt_rleq2,
rmul-rdiv-cancel2,
rmul-rdiv-cancel,
req_functionality,
rmul_preserves_req,
req_wf,
squash_wf,
true_wf,
iff_weakening_equal,
zero-rleq-rabs,
rmul-ac,
rmul_comm,
rmul_preserves_rleq2,
radd-zero-both,
radd-rminus-both,
rminus-rminus,
rmul-zero-both,
radd-int,
rmul_functionality,
rmul-distrib2,
rmul-identity1,
rminus-as-rmul,
radd-ac,
radd-assoc,
rminus-radd,
rmul-assoc,
rminus_functionality,
rmul_over_rminus,
rmul-distrib,
req_transitivity,
rabs-rmul,
req_inversion,
uimplies_transitivity,
radd-rminus-assoc,
radd_comm,
radd_functionality,
rabs_functionality,
req_weakening,
rleq_functionality,
uiff_transitivity,
r-triangle-inequality,
rminus_wf,
and_functionality_wrt_rev_uimplies,
rmin-rleq,
rmin_strict_ub,
rmin_wf,
radd_functionality_wrt_rleq,
radd_wf,
sq_stable__less_than,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
small-reciprocal-real,
sq_stable__rless,
less_than'_wf,
rsub_wf,
rmul_wf,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
r-bound-property,
mul_nat_plus,
less_than_wf,
r-bound_wf,
all_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_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_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
rleq-int-fractions2,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
continuous_functionality_wrt_subinterval,
i-approx_wf,
i-approx-is-subinterval,
Inorm-bound,
sq_stable__icompact,
icompact_wf,
rfun_subtype,
Inorm_wf,
uall_wf,
real_wf,
i-member_wf,
rleq_wf,
rabs_wf,
i-member-approx,
set_wf,
nat_plus_wf,
derivative_wf,
continuous_wf,
maps-compact_wf,
rfun_wf,
interval_wf
Rules used in proof :
lambdaEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
independent_isectElimination,
because_Cache,
applyEquality,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
introduction,
independent_functionElimination,
hypothesis,
rename,
setElimination,
isectElimination,
lemma_by_obid,
cut,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
unionElimination,
inrFormation,
multiplyEquality,
functionEquality,
independent_pairFormation,
natural_numberEquality,
productEquality,
axiomEquality,
minusEquality,
independent_pairEquality,
addEquality,
dependent_set_memberFormation,
equalityEquality,
addLevel,
impliesFunctionality,
isect_memberFormation,
universeEquality
Latex:
\mforall{}I,J:Interval. \mforall{}f,f':I {}\mrightarrow{}\mBbbR{}. \mforall{}g,g':J {}\mrightarrow{}\mBbbR{}.
(maps-compact(I;J;x.f[x])
{}\mRightarrow{} f[x] continuous for x \mmember{} I
{}\mRightarrow{} f'[x] continuous for x \mmember{} I
{}\mRightarrow{} g'[x] continuous for x \mmember{} J
{}\mRightarrow{} \mlambda{}x.f'[x] = d(f[x])/dx on I
{}\mRightarrow{} \mlambda{}x.g'[x] = d(g[x])/dx on J
{}\mRightarrow{} \mlambda{}x.g'[f[x]] * f'[x] = d(g[f[x]])/dx on I)
Date html generated:
2016_05_18-AM-10_15_09
Last ObjectModification:
2016_01_17-AM-00_57_38
Theory : reals
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