Nuprl Lemma : continuous_functionality_wrt_rfun-eq
∀I:Interval. ∀[f1,f2:I ⟶ℝ]. (rfun-eq(I;λx.f1[x];λx.f2[x]) ⇒ f1[x] continuous for x ∈ I ⇒ f2[x] continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun-eq: rfun-eq(I;f;g),
rfun: I ⟶ℝ,
interval: Interval,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
continuous: f[x] continuous for x ∈ I,
member: t ∈ T,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
rfun: I ⟶ℝ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
label: ...$L... t,
rfun-eq: rfun-eq(I;f;g),
r-ap: f(x),
uiff: uiff(P;Q)
Lemmas referenced :
req_weakening,
rsub_functionality,
rabs_functionality,
rleq_functionality,
interval_wf,
rfun_wf,
rfun-eq_wf,
continuous_wf,
icompact_wf,
nat_plus_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,
i-member-approx,
less_than_wf,
real_wf,
all_wf,
int-to-real_wf,
rless_wf,
i-approx_wf,
i-member_wf,
rsub_wf,
rabs_wf,
rleq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
introduction,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
promote_hyp,
independent_functionElimination,
lemma_by_obid,
isectElimination,
because_Cache,
productEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
independent_isectElimination,
inrFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality
Latex:
\mforall{}I:Interval
\mforall{}[f1,f2:I {}\mrightarrow{}\mBbbR{}].
(rfun-eq(I;\mlambda{}x.f1[x];\mlambda{}x.f2[x]) {}\mRightarrow{} f1[x] continuous for x \mmember{} I {}\mRightarrow{} f2[x] continuous for x \mmember{} I)
Date html generated:
2016_05_18-AM-09_09_32
Last ObjectModification:
2016_01_17-AM-02_37_03
Theory : reals
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