Nuprl Lemma : real-fun-iff-continuous
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. (real-fun(f;a;b) ⇐⇒ real-cont(f;a;b)) supposing a ≤ b
Proof
Definitions occuring in Statement :
real-cont: real-cont(f;a;b),
real-fun: real-fun(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
real-fun: real-fun(f;a;b),
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
subtype_rel: A ⊆r B,
so_apply: x[s],
continuous: f[x] continuous for x ∈ I,
i-approx: i-approx(I;n),
rccint: [l, u],
real-cont: real-cont(f;a;b),
exists: ∃x:A. B[x],
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+,
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,
rge: x ≥ y,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
real-continuity,
le_witness_for_triv,
real-fun_wf,
continuous-rneq,
rccint_wf,
subtype_rel_self,
rfun_wf,
req_wf,
i-member_wf,
real-cont_wf,
rleq_wf,
real_wf,
nat_plus_wf,
icompact_wf,
i-approx_wf,
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,
istype-void,
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,
req-iff-not-rneq,
rneq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isect_memberFormation_alt,
sqequalRule,
lambdaEquality_alt,
isectElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
independent_pairFormation,
universeIsType,
applyEquality,
independent_functionElimination,
setElimination,
setIsType,
dependent_set_memberFormation_alt,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
because_Cache,
closedConclusion,
inrFormation_alt,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
functionIsType
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. (real-fun(f;a;b) \mLeftarrow{}{}\mRightarrow{} real-cont(f;a;b)) supposing a \mleq{} b
Date html generated:
2019_10_30-AM-07_15_21
Last ObjectModification:
2019_10_09-PM-05_37_58
Theory : reals
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