Nuprl Lemma : max-metric-mdist-from-zero-2
∀[c:{c:ℝ| r0 ≤ c} ]. ∀[n:ℕ]. ∀[x:ℝ^n]. uiff(mdist(max-metric(n);λi.r0;x) ≤ c;∀i:ℕn. (x i ∈ [-(c), c]))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
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,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
real-vec: ℝ^n,
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
int-to-real_wf,
int_seg_wf,
real-vec_wf,
istype-nat,
real_wf,
rleq_wf,
le_witness_for_triv,
mdist_wf,
max-metric_wf,
iff_weakening_uiff,
i-member_wf,
rccint_wf,
rminus_wf,
max-metric-mdist-from-zero,
rleq_functionality,
mdist-symm,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
lambdaEquality_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
universeIsType,
natural_numberEquality,
hypothesisEquality,
setIsType,
independent_pairFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
functionEquality,
applyEquality,
independent_functionElimination,
promote_hyp,
functionIsType
Latex:
\mforall{}[c:\{c:\mBbbR{}| r0 \mleq{} c\} ]. \mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n].
uiff(mdist(max-metric(n);\mlambda{}i.r0;x) \mleq{} c;\mforall{}i:\mBbbN{}n. (x i \mmember{} [-(c), c]))
Date html generated:
2019_10_30-AM-08_36_21
Last ObjectModification:
2019_10_02-AM-11_02_18
Theory : reals
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