Nuprl Lemma : i-member-finite
∀[I:Interval]. ∀[r:ℝ]. left-endpoint(I)≤r≤right-endpoint(I) supposing r ∈ I supposing i-finite(I)
Proof
Definitions occuring in Statement :
i-member: r ∈ I,
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
i-finite: i-finite(I),
interval: Interval,
rbetween: x≤y≤z,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
interval: Interval,
i-member: r ∈ I,
i-finite: i-finite(I),
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
right-endpoint: right-endpoint(I),
left-endpoint: left-endpoint(I),
rbetween: x≤y≤z,
endpoints: endpoints(I),
outl: outl(x),
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
guard: {T},
bfalse: ff,
false: False,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ
Lemmas referenced :
rleq_weakening_rless,
less_than'_wf,
rsub_wf,
left-endpoint_wf,
real_wf,
nat_plus_wf,
right-endpoint_wf,
i-member_wf,
i-finite_wf,
interval_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
sqequalRule,
hypothesis,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
voidElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
setElimination,
rename,
minusEquality,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality
Latex:
\mforall{}[I:Interval]. \mforall{}[r:\mBbbR{}]. left-endpoint(I)\mleq{}r\mleq{}right-endpoint(I) supposing r \mmember{} I supposing i-finite(I)
Date html generated:
2016_05_18-AM-08_19_39
Last ObjectModification:
2015_12_27-PM-11_57_57
Theory : reals
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