Nuprl Lemma : real-ball-coordinate-range
∀[r:{r:ℝ| r0 ≤ r} ]. ∀[n:ℕ]. ∀[i:ℕn]. ∀[x:B(n;r)]. (x i ∈ [-(r), r])
Proof
Definitions occuring in Statement :
real-ball: B(n;r),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
real-ball: B(n;r),
real-vec: ℝ^n,
prop: ℙ,
implies: P ⇒ Q,
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
nat: ℕ,
rge: x ≥ y,
guard: {T},
iff: P ⇐⇒ Q
Lemmas referenced :
member_rccint_lemma,
istype-void,
sq_stable__and,
rleq_wf,
rminus_wf,
sq_stable__rleq,
le_witness_for_triv,
sq_stable__i-member,
rccint_wf,
real-ball_wf,
int_seg_wf,
istype-nat,
real_wf,
int-to-real_wf,
component-rleq-real-vec-norm,
rabs_wf,
real-vec-norm_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rabs-rleq-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
setElimination,
rename,
because_Cache,
applyEquality,
hypothesisEquality,
universeIsType,
independent_functionElimination,
lambdaFormation_alt,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
setIsType
Latex:
\mforall{}[r:\{r:\mBbbR{}| r0 \mleq{} r\} ]. \mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}n]. \mforall{}[x:B(n;r)]. (x i \mmember{} [-(r), r])
Date html generated:
2019_10_30-AM-10_14_53
Last ObjectModification:
2019_06_28-PM-01_52_11
Theory : real!vectors
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