Nuprl Lemma : interval-retraction_wf
∀[u,v,x:ℝ]. interval-retraction(u;v;x) ∈ {x:ℝ| x ∈ [u, v]} supposing u ≤ v
Proof
Definitions occuring in Statement :
interval-retraction: interval-retraction(u;v;r),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
interval-retraction: interval-retraction(u;v;r),
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
member_rccint_lemma,
rmin_wf,
rmax_wf,
rmin_ub,
rleq-rmax,
rmin-rleq,
rleq_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
dependent_set_memberEquality,
isectElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
independent_pairFormation,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache
Latex:
\mforall{}[u,v,x:\mBbbR{}]. interval-retraction(u;v;x) \mmember{} \{x:\mBbbR{}| x \mmember{} [u, v]\} supposing u \mleq{} v
Date html generated:
2017_10_03-AM-10_05_08
Last ObjectModification:
2017_07_10-PM-05_00_43
Theory : reals
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