Nuprl Lemma : mdist-max-metric-strict-lb
∀c:{c:ℝ| r0 < c} . ∀n:ℕ. ∀x,y:ℝ^n. ((∀i:ℕn. (|(x i) - y i| < c)) ⇒ (mdist(max-metric(n);x;y) < c))
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
max-metric: max-metric(n),
mdist: mdist(d;x;y),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_stable: SqStable(P),
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
nat: ℕ,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
real-vec: ℝ^n,
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+
Latex:
\mforall{}c:\{c:\mBbbR{}| r0 < c\} . \mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. ((\mforall{}i:\mBbbN{}n. (|(x i) - y i| < c)) {}\mRightarrow{} (mdist(max-metric(n);x;y) < c))
Date html generated:
2020_05_20-PM-00_42_47
Last ObjectModification:
2019_11_21-AM-06_09_21
Theory : reals
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