Nuprl Lemma : derivative-continuous
∀I:Interval. ∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . (g[x] ≠ g[y] ⇒ x ≠ y)) ⇒ λx.g[x] = d(f[x])/dx on I ⇒ g[x] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
derivative: λz.g[z] = d(f[x])/dx on I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
interval: Interval,
rneq: 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 :
rfun: I ⟶ℝ,
label: ...$L... t,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
continuous: f[x] continuous for x ∈ I,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
rless: x < y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
cand: A c∧ B,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
derivative: λz.g[z] = d(f[x])/dx on I,
subinterval: I ⊆ J ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rsub: x - y,
rgt: x > y
Lemmas referenced :
rmul_comm,
rless_functionality,
rmul_preserves_rless,
rless_transitivity1,
rneq-int,
equal_wf,
rless_irreflexivity,
rleq_weakening_rless,
rless_functionality_wrt_implies,
rless-int-fractions2,
rneq-iff-rabs,
rleq-iff-not-rless,
rminus-as-rmul,
radd-rminus-assoc,
radd_comm,
radd-ac,
radd-assoc,
rminus-rminus,
rminus-radd,
rminus_functionality,
rmul_over_rminus,
radd_functionality,
rmul-distrib,
req_transitivity,
rabs_functionality,
uiff_transitivity,
rabs-rmul,
rminus_wf,
r-triangle-inequality,
radd-int-fractions,
req_functionality,
multiply_nat_plus,
intformeq_wf,
int_formula_prop_eq_lemma,
req-int-fractions,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
rmul-distrib2,
req_inversion,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
radd_wf,
rmul_functionality,
rmul_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
req_weakening,
rabs-difference-symmetry,
rleq_functionality,
i-approx-is-subinterval,
mul_nat_plus,
less_than_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rless_wf,
int-to-real_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,
nat_plus_wf,
set_wf,
icompact_wf,
i-approx_wf,
derivative_wf,
real_wf,
i-member_wf,
all_wf,
rneq_wf,
rfun_wf,
interval_wf
Rules used in proof :
dependent_set_memberEquality,
functionEquality,
rename,
setElimination,
because_Cache,
setEquality,
applyEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
independent_functionElimination,
inrFormation,
independent_isectElimination,
productEquality,
productElimination,
baseClosed,
imageMemberEquality,
introduction,
independent_pairFormation,
natural_numberEquality,
dependent_functionElimination,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
equalityEquality,
addEquality,
minusEquality,
inlFormation
Latex:
\mforall{}I:Interval. \mforall{}f,g:I {}\mrightarrow{}\mBbbR{}.
((\mforall{}x,y:\{x:\mBbbR{}| x \mmember{} I\} . (g[x] \mneq{} g[y] {}\mRightarrow{} x \mneq{} y))
{}\mRightarrow{} \mlambda{}x.g[x] = d(f[x])/dx on I
{}\mRightarrow{} g[x] continuous for x \mmember{} I)
Date html generated:
2016_05_18-AM-10_00_54
Last ObjectModification:
2016_01_17-AM-00_44_47
Theory : reals
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