Nuprl Lemma : derivative-add
∀[I:Interval]. ∀[f1,f2,g1,g2:I ⟶ℝ].
(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.g1[x] + g2[x] on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
radd: a + b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
derivative: d(f[x])/dx = λz.g[z] on I,
all: ∀x:A. B[x],
member: t ∈ T,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P)
Lemmas referenced :
mul_nat_plus,
less_than_wf,
rmin_wf,
rmin_strict_ub,
i-member-approx,
rleq_wf,
rabs_wf,
rsub_wf,
i-member_wf,
i-approx_wf,
real_wf,
set_wf,
nat_plus_wf,
icompact_wf,
iproper_wf,
derivative_wf,
rfun_wf,
interval_wf,
rless_wf,
int-to-real_wf,
all_wf,
radd_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,
rmin-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
equal_wf,
req_wf,
rminus_wf,
req_weakening,
uiff_transitivity,
req_functionality,
radd_functionality,
rminus_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
rminus-radd,
req_inversion,
radd-assoc,
radd-ac,
radd_comm,
rmul_functionality,
rminus-as-rmul,
rminus-rminus,
rleq_functionality,
rabs_functionality,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_transitivity,
r-triangle-inequality,
radd_functionality_wrt_rleq,
rmul-distrib2,
radd-rdiv,
rdiv_functionality,
radd-int,
rleq-int-fractions,
sq_stable__less_than,
sq_stable__and,
sq_stable__icompact,
sq_stable__iproper,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rleq-int-fractions2,
zero-rleq-rabs,
rmul_functionality_wrt_rleq2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
setElimination,
rename,
dependent_set_memberFormation,
productElimination,
because_Cache,
independent_functionElimination,
lambdaEquality,
productEquality,
applyEquality,
setEquality,
functionEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
minusEquality,
multiplyEquality,
addEquality,
imageElimination,
inlFormation
Latex:
\mforall{}[I:Interval]. \mforall{}[f1,f2,g1,g2:I {}\mrightarrow{}\mBbbR{}].
(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.g1[x] + g2[x] on I)
Date html generated:
2017_10_03-PM-00_08_44
Last ObjectModification:
2017_07_28-AM-08_34_15
Theory : reals
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