Nuprl Lemma : max-metric-mdist-from-zero-strict
∀c:{c:ℝ| r0 < c} . ∀n:ℕ. ∀x:ℝ^n. (mdist(max-metric(n);x;λi.r0) < 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),
rooint: (l, u),
i-member: r ∈ I,
rless: x < y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n
Definitions unfolded in proof :
max-metric: max-metric(n),
mdist: mdist(d;x;y),
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
rless: x < y,
sq_exists: ∃x:A [B[x]],
false: False,
nat_plus: ℕ+,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
squash: ↓T,
nat: ℕ,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
real-vec: ℝ^n,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
req_int_terms: t1 ≡ t2,
real: ℝ
Latex:
\mforall{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}n:\mBbbN{}. \mforall{}x:\mBbbR{}\^{}n. (mdist(max-metric(n);x;\mlambda{}i.r0) < c \mLeftarrow{}{}\mRightarrow{} \mforall{}i:\mBbbN{}n. (x i \mmember{} (-(c), c)))
Date html generated:
2020_05_20-PM-00_43_39
Last ObjectModification:
2019_11_21-AM-06_35_35
Theory : reals
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