Nuprl Lemma : rleq_inf
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. (inf(A) = b ⇒ (c ≤ b ⇐⇒ ∀x:ℝ. ((x ∈ A) ⇒ (c ≤ x))))
Proof
Definitions occuring in Statement :
inf: inf(A) = b,
rset-member: x ∈ A,
rset: Set(ℝ),
rleq: x ≤ y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
inf: inf(A) = b,
prop: ℙ,
rev_implies: P ⇐ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uimplies: b supposing a,
lower-bound: lower-bound(A;b),
guard: {T},
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x]
Lemmas referenced :
rset-member_wf,
rleq_wf,
inf_wf,
le_witness_for_triv,
real_wf,
rset_wf,
rleq_transitivity,
rleq-iff-all-rless,
rless_wf,
int-to-real_wf,
sq_stable__rless,
rless_transitivity2,
radd_wf,
rleq_weakening_rless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
functionIsType,
lambdaEquality_alt,
dependent_functionElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_functionElimination,
setIsType,
natural_numberEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[A:Set(\mBbbR{})]. \mforall{}[b,c:\mBbbR{}]. (inf(A) = b {}\mRightarrow{} (c \mleq{} b \mLeftarrow{}{}\mRightarrow{} \mforall{}x:\mBbbR{}. ((x \mmember{} A) {}\mRightarrow{} (c \mleq{} x))))
Date html generated:
2019_10_29-AM-10_40_00
Last ObjectModification:
2019_04_17-PM-03_31_31
Theory : reals
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