Nuprl Lemma : Cauchy-Schwarz-strict
∀n:ℕ. ∀x,y:ℝ^n. (∃i,j:ℕn. (x j) * (y i) ≠ (x i) * (y j) ⇐⇒ |x⋅y| < (||x|| * ||y||))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec: ℝ^n,
rneq: x ≠ y,
rless: x < y,
rabs: |x|,
rmul: a * b,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
real-vec: ℝ^n,
so_apply: x[s],
dot-product: x⋅y,
real-vec-norm: ||x||,
uall: ∀[x:A]. B[x]
Lemmas referenced :
Cauchy-Schwarz3-strict,
real-vec_wf,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
isectElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. (\mexists{}i,j:\mBbbN{}n. (x j) * (y i) \mneq{} (x i) * (y j) \mLeftarrow{}{}\mRightarrow{} |x\mcdot{}y| < (||x|| * ||y||))
Date html generated:
2017_10_03-AM-10_52_41
Last ObjectModification:
2017_06_19-PM-04_14_00
Theory : reals
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