Nuprl Lemma : continuous-abs-subtype
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⊆r |f[x]| continuous for x ∈ I)
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
interval: Interval,
rabs: |x|,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
continuous-abs-ext,
implies: P ⇒ Q,
all: ∀x:A. B[x],
continuous: f[x] continuous for x ∈ I,
sq_exists: ∃x:{A| B[x]},
and: P ∧ Q,
nat_plus: ℕ+,
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,
squash: ↓T,
true: True,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B
Lemmas referenced :
continuous_wf,
i-member_wf,
real_wf,
rfun_wf,
interval_wf,
continuous-abs-ext,
isect_wf,
rabs_wf,
equal_wf,
nat_plus_wf,
set_wf,
icompact_wf,
i-approx_wf,
rless_wf,
int-to-real_wf,
less_than_wf,
rleq_wf,
rsub_wf,
i-member-approx,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
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,
squash_wf,
true_wf,
iff_weakening_equal,
subtype_rel_dep_function,
rmin_wf,
rmin-idempotent-eq,
sq_exists_wf,
less_than'_wf,
sq_stable__and,
sq_stable__rless,
sq_stable__all,
sq_stable__rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
applyEquality,
setElimination,
dependent_set_memberEquality,
hypothesis,
setEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
instantiate,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
isectEquality,
functionExtensionality,
productEquality,
natural_numberEquality,
independent_isectElimination,
inrFormation,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
minusEquality,
independent_pairEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[f:I {}\mrightarrow{}\mBbbR{}]. (f[x] continuous for x \mmember{} I \msubseteq{}r |f[x]| continuous for x \mmember{} I)
Date html generated:
2017_10_03-AM-10_23_04
Last ObjectModification:
2017_07_28-AM-08_07_28
Theory : reals
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