Nuprl Lemma : i-finite-iff-bounded
∀I:Interval. (i-finite(I) ⇐⇒ ∃a,b:ℝ. ∀[r:ℝ]. a≤r≤b supposing r ∈ I)
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
i-finite: i-finite(I),
interval: Interval,
rbetween: x≤y≤z,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
rbetween: x≤y≤z,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
real: ℝ,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
guard: {T},
rsub: x - y,
interval: Interval,
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cand: A c∧ B,
bfalse: ff,
i-member: r ∈ I,
rge: x ≥ y,
or: P ∨ Q
Lemmas referenced :
i-finite_wf,
exists_wf,
real_wf,
uall_wf,
isect_wf,
i-member_wf,
rbetween_wf,
interval_wf,
left-endpoint_wf,
right-endpoint_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
i-member-finite,
rleq_wf,
radd_wf,
int-to-real_wf,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
uiff_transitivity,
rleq_functionality,
req_transitivity,
radd_functionality,
req_weakening,
rminus-as-rmul,
radd-assoc,
req_inversion,
rmul-identity1,
rmul-distrib2,
rmul_functionality,
radd-int,
rmul-zero-both,
radd-zero-both,
rless-int,
rless_transitivity1,
rless_irreflexivity,
uiff_transitivity2,
radd-ac,
radd_comm,
radd-rminus-both,
squash_wf,
true_wf,
rminus-int,
rmax_wf,
rleq-rmax,
rmax_lb,
trivial-rless-radd,
rless_functionality_wrt_implies,
rleq_weakening_equal,
rmin_wf,
rmin-rleq,
rmin_ub,
rmin_strict_lb,
trivial-rsub-rless,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
isect_memberFormation,
productElimination,
independent_pairEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
addEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
inlFormation
Latex:
\mforall{}I:Interval. (i-finite(I) \mLeftarrow{}{}\mRightarrow{} \mexists{}a,b:\mBbbR{}. \mforall{}[r:\mBbbR{}]. a\mleq{}r\mleq{}b supposing r \mmember{} I)
Date html generated:
2016_10_26-AM-09_29_03
Last ObjectModification:
2016_09_11-PM-07_52_21
Theory : reals
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