Nuprl Lemma : rabs-nonzero-on-compact
∀a,b:ℝ.
((a ≤ b)
⇒ (∀f:[a, b] ⟶ℝ. ∀k:ℕ+.
(f[x] continuous for x ∈ [a, b]
⇒ (∀x:ℝ. ((x ∈ [a, b]) ⇒ ((r1/r(k)) ≤ |f[x]|)))
⇒ ((∀x:ℝ. ((x ∈ [a, b]) ⇒ ((r1/r(k)) ≤ f[x]))) ∨ (∀x:ℝ. ((x ∈ [a, b]) ⇒ (f[x] ≤ (r(-1)/r(k)))))))))
Proof
Definitions occuring in Statement :
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
rless: x < y,
sq_exists: ∃x:A [B[x]],
r-ap: f(x),
less_than: a < b,
squash: ↓T,
true: True,
le: A ≤ B,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
subtype_rel: A ⊆r B,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
i-member: r ∈ I,
rccint: [l, u],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
so_apply: x[s],
rfun: I ⟶ℝ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
label: ...$L... t
Lemmas referenced :
r-ap_wf,
rless_irreflexivity,
rabs_functionality,
sq_stable__i-member,
rmin_ub,
rmax_lb,
rleq_transitivity,
rmin_wf,
rmax_wf,
mul_nat_plus,
less_than_wf,
rmul_preserves_rless,
rless-int-fractions,
intermediate-value-theorem,
rless_transitivity2,
rleq_weakening_rless,
rless_transitivity1,
rless_functionality,
req_transitivity,
rmul-rinv,
rmul_reverses_rleq,
rminus_wf,
rleq-int,
false_wf,
less_than'_wf,
rsub_wf,
rmul_wf,
rinv_wf2,
uiff_transitivity,
rleq_functionality,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_weakening,
rminus_functionality,
rinv-as-rdiv,
rleq_weakening_equal,
rabs-ub,
rdiv_wf,
int-to-real_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,
rless_wf,
rless-int-fractions2,
itermMultiply_wf,
int_term_value_mul_lemma,
member_rccint_lemma,
rleq_wf,
i-member_wf,
rccint_wf,
all_wf,
real_wf,
rabs_wf,
continuous_wf,
nat_plus_wf,
rfun_wf,
rminus-rdiv,
rmul-one-both,
rmul_over_rminus,
rmul-minus,
rmul_reverses_rleq_iff,
squash_wf,
true_wf,
rneq_wf,
rminus-int,
iff_weakening_equal
Rules used in proof :
universeEquality,
lemma_by_obid,
promote_hyp,
imageElimination,
imageMemberEquality,
baseClosed,
addLevel,
isect_memberFormation,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
inrFormation,
productElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
applyEquality,
dependent_set_memberEquality,
productEquality,
inlFormation,
functionEquality,
minusEquality,
setEquality
Latex:
\mforall{}a,b:\mBbbR{}.
((a \mleq{} b)
{}\mRightarrow{} (\mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}. \mforall{}k:\mBbbN{}\msupplus{}.
(f[x] continuous for x \mmember{} [a, b]
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} ((r1/r(k)) \mleq{} |f[x]|)))
{}\mRightarrow{} ((\mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} ((r1/r(k)) \mleq{} f[x])))
\mvee{} (\mforall{}x:\mBbbR{}. ((x \mmember{} [a, b]) {}\mRightarrow{} (f[x] \mleq{} (r(-1)/r(k)))))))))
Date html generated:
2018_05_22-PM-02_46_47
Last ObjectModification:
2018_05_20-PM-02_47_16
Theory : reals
Home
Index