Nuprl Lemma : differentiable-continuous
∀I:Interval. ∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (g[x] = g[y]))) ⇒ d(f[x])/dx = λx.g[x] on I ⇒ f[x] (proper)continuous for x ∈ 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,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
req: x = y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]}
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,
proper-continuous: f[x] (proper)continuous for x ∈ I,
nat_plus: ℕ+,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
subinterval: I ⊆ J ,
sq_stable: SqStable(P),
squash: ↓T,
r-ap: f(x),
exists: ∃x:A. B[x],
sup: sup(A) = b,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
rge: x ≥ y,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
cand: A c∧ B,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
upper-bound: A ≤ b,
derivative: d(f[x])/dx = λz.g[z] on I,
less_than: a < b,
sq_exists: ∃x:A [B[x]],
rleq: x ≤ y,
rnonneg: rnonneg(x),
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rless: x < y,
nequal: a ≠ b ∈ T
Lemmas referenced :
function-proper-continuous,
i-member_wf,
real_wf,
derivative_wf,
all_wf,
req_wf,
rfun_wf,
interval_wf,
i-approx-is-subinterval,
less_than_wf,
sup-range,
i-approx_wf,
icompact_wf,
rabs_wf,
subtype_rel_sets,
continuous-abs,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
proper-continuous-implies,
sq_stable__icompact,
sq_stable__iproper,
r-archimedean,
upper-bound_functionality,
rrange_wf,
int-to-real_wf,
upper-bound_wf,
nat_plus_wf,
iproper_wf,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
rleq-int,
nat_properties,
nat_plus_properties,
sq_stable__and,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rset-member-rrange,
rmul_wf,
rsub_wf,
zero-rleq-rabs,
rleq_wf,
rleq_weakening_equal,
rleq_functionality,
rabs-rmul,
req_weakening,
rleq_functionality_wrt_implies,
rmul_functionality_wrt_rleq2,
r-triangle-inequality,
less_than'_wf,
radd_wf,
rdiv_wf,
rless-int,
rless_wf,
equal_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
rinv_wf2,
itermMultiply_wf,
rleq_weakening,
radd_functionality_wrt_rleq,
rmul_functionality,
rabs-difference-symmetry,
rabs_functionality,
req_transitivity,
radd_functionality,
rinv1,
rmul-identity1,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
req_inversion,
radd-int,
rmul-distrib2,
rmin_wf,
mul_bounds_1b,
less-iff-le,
rmin_strict_ub,
rless-int-fractions2,
mul_nat_plus,
intformless_wf,
int_formula_prop_less_lemma,
int_term_value_mul_lemma,
intformand_wf,
int_formula_prop_and_lemma,
rmin_ub,
rneq_functionality,
rmul-int,
rneq-int,
int_entire_a,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
rdiv_functionality,
squash_wf,
true_wf,
iff_weakening_equal,
rmul-neq-zero,
rleq-implies-rleq,
rmul_preserves_rleq2,
rleq_weakening_rless,
rmul-one,
rinv-of-rmul,
rmul-rinv,
rinv-as-rdiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
hypothesis,
isectElimination,
setEquality,
independent_functionElimination,
because_Cache,
functionEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
productEquality,
addEquality,
unionElimination,
independent_pairFormation,
voidElimination,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
approximateComputation,
int_eqEquality,
inlFormation,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
axiomEquality,
inrFormation,
dependent_set_memberFormation,
multiplyEquality,
baseApply,
closedConclusion,
universeEquality
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . ((x = y) {}\mRightarrow{} (g[x] = g[y])))
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.g[x] on I
{}\mRightarrow{} f[x] (proper)continuous for x \mmember{} I)
Date html generated:
2018_05_22-PM-02_44_44
Last ObjectModification:
2017_10_21-PM-07_20_16
Theory : reals
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