Nuprl Lemma : Cauchy-Schwarz-equality2
∀n:ℕ. ∀x,y:ℝ^n. (¬(|x⋅y| < (||x|| * ||y||)) ⇐⇒ (r0 < ||y||) ⇒ (∃t:ℝ. req-vec(n;x;t*y)))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-mul: a*X,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T},
uimplies: b supposing a,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
or: P ∨ Q,
rneq: x ≠ y,
stable: Stable{P},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
less_than: a < b,
ge: i ≥ j ,
nat_plus: ℕ+,
sq_exists: ∃x:A [B[x]],
rless: x < y,
req_int_terms: t1 ≡ t2
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. (\mneg{}(|x\mcdot{}y| < (||x|| * ||y||)) \mLeftarrow{}{}\mRightarrow{} (r0 < ||y||) {}\mRightarrow{} (\mexists{}t:\mBbbR{}. req-vec(n;x;t*y)))
Date html generated:
2020_05_20-PM-00_41_30
Last ObjectModification:
2020_01_06-PM-00_16_20
Theory : reals
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