Nuprl Lemma : real-vec-norm-eq-iff
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[r:ℝ]. uiff(||x|| = r;(x⋅x = r^2) ∧ (r0 ≤ r))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
rleq: x ≤ y,
rnexp: x^k1,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
rleq: x ≤ y,
rnonneg: rnonneg(x),
all: ∀x:A. B[x],
prop: ℙ,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
real-vec-norm: ||x||,
subtype_rel: A ⊆r B
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbR{}\^{}n]. \mforall{}[r:\mBbbR{}]. uiff(||x|| = r;(x\mcdot{}x = r\^{}2) \mwedge{} (r0 \mleq{} r))
Date html generated:
2020_05_20-PM-00_36_16
Last ObjectModification:
2019_12_14-PM-03_05_00
Theory : reals
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