Nuprl Lemma : derivative_unique
∀[I:Interval]
(iproper(I) ⇒ (∀[f,g1,g2:I ⟶ℝ]. (d(f[x])/dx = λx.g1[x] on I ⇒ d(f[x])/dx = λx.g2[x] on I ⇒ rfun-eq(I;g1;g2))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun-eq: rfun-eq(I;f;g),
rfun: I ⟶ℝ,
iproper: iproper(I),
interval: Interval,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
rfun-eq: rfun-eq(I;f;g),
all: ∀x:A. B[x],
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
derivative: d(f[x])/dx = λz.g[z] on I,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
sq_exists: ∃x:{A| B[x]},
rless: x < y,
cand: A c∧ B,
subinterval: I ⊆ J ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
r-ap: f(x),
rdiv: (x/y),
subtype_rel: A ⊆r B
Lemmas referenced :
req-iff-rabs-rleq,
r-ap_wf,
sq_stable__i-member,
nat_plus_wf,
set_wf,
real_wf,
i-member_wf,
derivative_wf,
req_witness,
rfun_wf,
iproper_wf,
interval_wf,
sq_stable__rleq,
rabs_wf,
rsub_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,
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,
rless_wf,
i-member-proper-iff,
i-approx-compact,
mul_nat_plus,
less_than_wf,
icompact_wf,
i-approx_wf,
compact-proper-interval-near-member,
rleq_wf,
rmin_strict_ub,
rmin_wf,
i-approx-is-subinterval,
rmin_ub,
rleq_functionality_wrt_implies,
rmul_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_weakening_equal,
rleq_weakening,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
radd_wf,
equal_wf,
rminus_wf,
radd_functionality_wrt_rleq,
r-triangle-inequality,
req_functionality,
req_inversion,
rabs-rmul,
rabs_functionality,
itermAdd_wf,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rmul-distrib1,
req_wf,
rneq_functionality,
rmul-int,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
rinv_wf2,
uiff_transitivity,
rmul_functionality,
rdiv_functionality,
rinv-of-rmul,
req_transitivity,
rmul-rinv3,
rinv-mul-as-rdiv,
rmul_preserves_rleq,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
because_Cache,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
setEquality,
isect_memberEquality,
natural_numberEquality,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addLevel,
levelHypothesis,
productEquality,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
impliesFunctionality,
universeEquality
Latex:
\mforall{}[I:Interval]
(iproper(I)
{}\mRightarrow{} (\mforall{}[f,g1,g2:I {}\mrightarrow{}\mBbbR{}].
(d(f[x])/dx = \mlambda{}x.g1[x] on I {}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g2[x] on I {}\mRightarrow{} rfun-eq(I;g1;g2))))
Date html generated:
2017_10_03-PM-00_07_34
Last ObjectModification:
2017_07_28-AM-08_33_22
Theory : reals
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