Nuprl Lemma : cantor-lemma
∀x,y,e,z:ℝ.
(∃x',y':ℝ. ((x ≤ x') ∧ (x' < y') ∧ (y' ≤ y) ∧ ((z < x') ∨ (y' < z)) ∧ ((y' - x') < e))) supposing
((x < y) and
(r0 < e))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
guard: {T},
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
rsub: x - y
Lemmas referenced :
rless-cases,
sq_stable__rless,
rless_wf,
int-to-real_wf,
real_wf,
ravg-between,
rmin_wf,
rmin_strict_ub,
ravg_wf,
equal_wf,
radd_wf,
trivial-rless-radd,
rleq_weakening_rless,
radd-preserves-rless,
rsub_wf,
rleq_wf,
or_wf,
exists_wf,
rminus_wf,
rless_functionality,
radd-rminus-assoc,
req_weakening,
radd_functionality,
radd_comm,
rmax_wf,
rmax_strict_lb,
trivial-rsub-rless,
radd-zero-both,
radd-rminus-both,
req_transitivity,
radd-ac
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
because_Cache,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
isectElimination,
natural_numberEquality,
productElimination,
independent_pairFormation,
rename,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
addLevel,
levelHypothesis,
promote_hyp,
inrFormation,
lambdaEquality,
inlFormation
Latex:
\mforall{}x,y,e,z:\mBbbR{}.
(\mexists{}x',y':\mBbbR{}. ((x \mleq{} x') \mwedge{} (x' < y') \mwedge{} (y' \mleq{} y) \mwedge{} ((z < x') \mvee{} (y' < z)) \mwedge{} ((y' - x') < e))) supposing
((x < y) and
(r0 < e))
Date html generated:
2017_10_03-AM-09_11_50
Last ObjectModification:
2017_07_28-AM-07_42_45
Theory : reals
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