Nuprl Lemma : real-cont_wf
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. (real-cont(f;a;b) ∈ ℙ)
Proof
Definitions occuring in Statement :
real-cont: real-cont(f;a;b),
rfun: I ⟶ℝ,
rccint: [l, u],
real: ℝ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
sq_exists: ∃x:{A| B[x]},
rless: x < y,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
nat_plus: ℕ+,
rfun: I ⟶ℝ,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
top: Top,
all: ∀x:A. B[x],
real-cont: real-cont(f;a;b),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rccint_wf,
rfun_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
rdiv_wf,
rsub_wf,
rabs_wf,
rleq_wf,
int-to-real_wf,
rless_wf,
real_wf,
exists_wf,
nat_plus_wf,
all_wf,
member_rccint_lemma
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
computeAll,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
independent_functionElimination,
inrFormation,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
applyEquality,
functionEquality,
productEquality,
because_Cache,
rename,
setElimination,
lambdaFormation,
hypothesisEquality,
natural_numberEquality,
setEquality,
lambdaEquality,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[f:[a, b] {}\mrightarrow{}\mBbbR{}]. (real-cont(f;a;b) \mmember{} \mBbbP{})
Date html generated:
2016_07_08-PM-06_03_26
Last ObjectModification:
2016_07_05-PM-02_47_18
Theory : reals
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