Nuprl Lemma : derivative-mul-part1
∀I:Interval
(True
⇒ (∀f1,f2,g1,g2:I ⟶ℝ.
(f1(x) (proper)continuous for x ∈ I
⇒ f2(x) (proper)continuous for x ∈ I
⇒ g2(x) (proper)continuous for x ∈ I
⇒ d(f1[x])/dx = λx.g1[x] on I
⇒ d(f2[x])/dx = λx.g2[x] on I
⇒ d(f1[x] * f2[x])/dx = λx.(f1[x] * g2[x]) + (f2[x] * g1[x]) on I)))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
proper-continuous: f[x] (proper)continuous for x ∈ I,
r-ap: f(x),
rfun: I ⟶ℝ,
interval: Interval,
rmul: a * b,
radd: a + b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
true: True
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
derivative: d(f[x])/dx = λz.g[z] on I,
member: t ∈ T,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
exists: ∃x:A. B[x],
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
cand: A c∧ B,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
rge: x ≥ y,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:{A| B[x]},
proper-continuous: f[x] (proper)continuous for x ∈ I,
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
r-ap: f(x),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
rdiv: (x/y),
sq_type: SQType(T),
real: ℝ
Lemmas referenced :
i-approx-is-subinterval,
less_than_wf,
set_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
iproper_wf,
derivative_wf,
i-member_wf,
real_wf,
proper-continuous_wf,
r-ap_wf,
sq_stable__i-member,
rfun_wf,
true_wf,
interval_wf,
sq_stable__and,
sq_stable__icompact,
sq_stable__iproper,
rabs_wf,
rleq_wf,
uall_wf,
Inorm_wf,
Inorm-bound,
proper-continuous-implies,
rmax_functionality,
rmax-int,
req_inversion,
req_transitivity,
req_weakening,
rleq_functionality,
rmax_ub,
rmax_wf,
int-to-real_wf,
all_wf,
equal_wf,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
imax_nat_plus,
r-bound_wf,
imax_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
r-bound-property,
mul_nat_plus,
rmin_wf,
rsub_wf,
rless_wf,
rmul_wf,
radd_wf,
rdiv_wf,
rless-int,
rmin_strict_ub,
rminus_wf,
uimplies_transitivity,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rleq_weakening,
radd_functionality,
rabs-rmul,
rmul_functionality,
rabs_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
req-iff-rsub-is-0,
rmin-rleq,
rmin_lb,
multiply_nat_plus,
int_term_value_mul_lemma,
rmul_preserves_rleq2,
zero-rleq-rabs,
less_than'_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
int_entire_a,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
rinv_functionality2,
rinv-of-rmul,
rinv-mul-as-rdiv,
rdiv_functionality,
mul_bounds_1b,
rleq_weakening_rless,
subtype_base_sq,
int_term_value_add_lemma,
rmul-rinv,
rmul-rinv3,
rmul_preserves_rleq,
radd-preserves-rleq,
rmul_functionality_wrt_rleq2,
rmul-nonneg-case1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
setElimination,
rename,
hypothesis,
isectElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
setEquality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
productElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
unionElimination,
applyLambdaEquality,
inrFormation,
inlFormation,
dependent_set_memberFormation,
functionEquality,
multiplyEquality,
isect_memberFormation,
independent_pairEquality,
minusEquality,
axiomEquality,
baseApply,
closedConclusion,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}I:Interval
(True
{}\mRightarrow{} (\mforall{}f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}.
(f1(x) (proper)continuous for x \mmember{} I
{}\mRightarrow{} f2(x) (proper)continuous for x \mmember{} I
{}\mRightarrow{} g2(x) (proper)continuous for x \mmember{} I
{}\mRightarrow{} d(f1[x])/dx = \mlambda{}x.g1[x] on I
{}\mRightarrow{} d(f2[x])/dx = \mlambda{}x.g2[x] on I
{}\mRightarrow{} d(f1[x] * f2[x])/dx = \mlambda{}x.(f1[x] * g2[x]) + (f2[x] * g1[x]) on I)))
Date html generated:
2017_10_03-PM-00_10_29
Last ObjectModification:
2017_07_28-AM-08_34_50
Theory : reals
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