Nuprl Lemma : inf-rleq
∀[A:Set(ℝ)]. ∀b,c:ℝ. (inf(A) = b ⇒ (b ≤ c ⇐⇒ ∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ A) ∧ ((x - e) ≤ c))))))
Proof
Definitions occuring in Statement :
inf: inf(A) = b,
rset-member: x ∈ A,
rset: Set(ℝ),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
inf: inf(A) = b,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uimplies: b supposing a,
exists: ∃x:A. B[x],
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
lower-bound: lower-bound(A;b),
rge: x ≥ y
Lemmas referenced :
rless_wf,
int-to-real_wf,
rleq_wf,
le_witness_for_triv,
rset-member_wf,
rsub_wf,
inf_wf,
real_wf,
rset_wf,
radd-preserves-rless,
radd_wf,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
itermVar_wf,
rless_functionality,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rless_transitivity1,
rleq_weakening_rless,
rleq_functionality,
rleq-iff-all-rless,
sq_stable__rleq,
rleq-implies-rleq,
rleq_functionality_wrt_implies,
rleq_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
inhabitedIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
functionIsType,
productIsType,
because_Cache,
independent_functionElimination,
dependent_pairFormation_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
setIsType,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[A:Set(\mBbbR{})]
\mforall{}b,c:\mBbbR{}. (inf(A) = b {}\mRightarrow{} (b \mleq{} c \mLeftarrow{}{}\mRightarrow{} \mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}x:\mBbbR{}. ((x \mmember{} A) \mwedge{} ((x - e) \mleq{} c))))))
Date html generated:
2019_10_29-AM-10_40_17
Last ObjectModification:
2019_04_19-PM-03_59_11
Theory : reals
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