Nuprl Lemma : interior-dense-in-interval
∀a,b:ℝ. dense-in-interval([a, b];λr.((a < r) ∧ (r < b)))
Proof
Definitions occuring in Statement :
dense-in-interval: dense-in-interval(I;X),
rccint: [l, u],
rless: x < y,
real: ℝ,
all: ∀x:A. B[x],
and: P ∧ Q,
lambda: λx.A[x]
Definitions unfolded in proof :
so_apply: x[s],
and: P ∧ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
top: Top,
member: t ∈ T,
dense-in-interval: dense-in-interval(I;X),
all: ∀x:A. B[x],
squash: ↓T,
uimplies: b supposing a,
guard: {T},
sq_stable: SqStable(P),
cand: A c∧ B,
exists: ∃x:A. B[x]
Lemmas referenced :
rleq_wf,
real_wf,
set_wf,
rless_wf,
member_rccint_lemma,
rless_transitivity1,
rless_transitivity2,
sq_stable__rless,
ravg_wf,
ravg-between
Rules used in proof :
productEquality,
lambdaEquality,
hypothesisEquality,
rename,
setElimination,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
imageElimination,
baseClosed,
imageMemberEquality,
independent_isectElimination,
productElimination,
because_Cache,
dependent_pairFormation,
independent_functionElimination
Latex:
\mforall{}a,b:\mBbbR{}. dense-in-interval([a, b];\mlambda{}r.((a < r) \mwedge{} (r < b)))
Date html generated:
2017_10_03-AM-10_20_21
Last ObjectModification:
2017_07_31-PM-00_21_42
Theory : reals
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