Nuprl Lemma : proper-continuous-implies-functional
∀I:Interval. ∀f:I ⟶ℝ.
(f[x] (proper)continuous for x ∈ I ⇒ iproper(I) ⇒ (∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))))
Proof
Definitions occuring in Statement :
proper-continuous: f[x] (proper)continuous for x ∈ I,
rfun: I ⟶ℝ,
i-member: r ∈ I,
iproper: iproper(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,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
rfun: I ⟶ℝ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
cand: A c∧ B,
proper-continuous: f[x] (proper)continuous for x ∈ I,
rev_implies: P ⇐ Q,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rless: x < y,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_wf,
set_wf,
real_wf,
i-member_wf,
iproper_wf,
proper-continuous_wf,
rfun_wf,
interval_wf,
i-member-proper-iff,
sq_stable__i-member,
icompact_wf,
i-approx_wf,
i-approx-compact,
req-iff-rabs-rleq,
sq_stable__rleq,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
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,
nat_plus_wf,
itermSubtract_wf,
req-iff-rsub-is-0,
minus-zero,
rleq_weakening_rless,
i-member_functionality,
req_inversion,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_transitivity,
rabs-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
setEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality,
because_Cache,
natural_numberEquality,
independent_isectElimination,
inrFormation,
unionElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality
Latex:
\mforall{}I:Interval. \mforall{}f:I {}\mrightarrow{}\mBbbR{}.
(f[x] (proper)continuous for x \mmember{} I
{}\mRightarrow{} iproper(I)
{}\mRightarrow{} (\mforall{}a,b:\{x:\mBbbR{}| x \mmember{} I\} . ((a = b) {}\mRightarrow{} (f[a] = f[b]))))
Date html generated:
2018_05_22-PM-02_17_18
Last ObjectModification:
2017_10_21-PM-07_38_10
Theory : reals
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