Nuprl Lemma : real-cont-iff-continuous
∀a,b:ℝ. ((a ≤ b) ⇒ (∀f:[a, b] ⟶ℝ. (real-cont(f;a;b) ⇐⇒ f[x] continuous for x ∈ [a, b])))
Proof
Definitions occuring in Statement :
real-cont: real-cont(f;a;b),
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
real-cont: real-cont(f;a;b),
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
exists: ∃x:A. B[x],
i-approx: i-approx(I;n),
rccint: [l, u],
top: Top,
sq_exists: ∃x:A [B[x]],
cand: A c∧ B,
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
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,
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B
Lemmas referenced :
nat_plus_wf,
icompact_wf,
i-approx_wf,
rccint_wf,
real-cont_wf,
continuous_wf,
i-member_wf,
rfun_wf,
rleq_wf,
real_wf,
member_rccint_lemma,
istype-void,
sq_stable__rless,
int-to-real_wf,
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
istype-less_than,
rccint-icompact,
sq_stable__rleq,
le_witness_for_triv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
setIsType,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
dependent_functionElimination,
productElimination,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberFormation_alt,
setElimination,
rename,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
dependent_set_memberEquality_alt,
productIsType,
closedConclusion,
independent_isectElimination,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
functionIsTypeImplies
Latex:
\mforall{}a,b:\mBbbR{}. ((a \mleq{} b) {}\mRightarrow{} (\mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. (real-cont(f;a;b) \mLeftarrow{}{}\mRightarrow{} f[x] continuous for x \mmember{} [a, b])))
Date html generated:
2019_10_30-AM-07_15_43
Last ObjectModification:
2019_10_09-PM-06_37_59
Theory : reals
Home
Index