Nuprl Lemma : interval-retraction-req
∀[u,v:ℝ]. ∀[x:{x:ℝ| x ∈ [u, v]} ]. (interval-retraction(u;v;x) = x)
Proof
Definitions occuring in Statement :
interval-retraction: interval-retraction(u;v;r),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
interval-retraction: interval-retraction(u;v;r),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
real: ℝ,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member_rccint_lemma,
rleq_transitivity,
set_wf,
real_wf,
i-member_wf,
rccint_wf,
rleq_wf,
less_than'_wf,
rsub_wf,
nat_plus_wf,
squash_wf,
sq_stable__rleq,
sq_stable__and,
rmax-req,
rmin-req,
req_wf,
rmin_wf,
rmax_wf,
req_weakening,
uiff_transitivity,
req_functionality,
rmin_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productElimination,
isectElimination,
because_Cache,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
applyEquality,
minusEquality,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_pairFormation,
lambdaFormation,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[u,v:\mBbbR{}]. \mforall{}[x:\{x:\mBbbR{}| x \mmember{} [u, v]\} ]. (interval-retraction(u;v;x) = x)
Date html generated:
2017_10_03-AM-10_05_39
Last ObjectModification:
2017_07_10-PM-05_04_39
Theory : reals
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