Nuprl Lemma : inf-as-sup
∀[A:Set(ℝ)]. ∀b:ℝ. (inf(A) = b ⇐⇒ sup(-(A)) = -(b))
Proof
Definitions occuring in Statement :
rset-neg: -(A),
inf: inf(A) = b,
sup: sup(A) = b,
rset: Set(ℝ),
rminus: -(x),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
sup: sup(A) = b,
inf: inf(A) = b,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
upper-bound: A ≤ b,
lower-bound: lower-bound(A;b),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
rsub: x - y,
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A
Lemmas referenced :
member_rset_neg_lemma,
istype-void,
rmul_reverses_rleq_iff,
int-to-real_wf,
rminus_wf,
rless-int,
le_witness_for_triv,
rset-member_wf,
squash_wf,
true_wf,
rminus-rminus-eq,
subtype_rel_self,
iff_weakening_equal,
rmul_reverses_rless_iff,
rsub_wf,
rless_wf,
rleq_wf,
radd_wf,
real_wf,
rset_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
itermAdd_wf,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_functionality,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
productElimination,
isectElimination,
hypothesisEquality,
minusEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
dependent_pairFormation_alt,
promote_hyp,
applyEquality,
imageElimination,
because_Cache,
instantiate,
universeEquality,
productIsType,
functionIsType,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[A:Set(\mBbbR{})]. \mforall{}b:\mBbbR{}. (inf(A) = b \mLeftarrow{}{}\mRightarrow{} sup(-(A)) = -(b))
Date html generated:
2019_10_29-AM-10_44_27
Last ObjectModification:
2019_04_19-PM-06_09_11
Theory : reals
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