Nuprl Lemma : Cauchy-Schwarz-non-equality1
∀n:ℕ. ∀x,y:ℝ^n. ((r0 < ||y||) ⇒ (∀a:ℝ. x ≠ a*y) ⇒ (|x⋅y| < (||x|| * ||y||)))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-mul: a*X,
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],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
real-vec: ℝ^n,
uimplies: b supposing a,
nat: ℕ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
real-vec-mul: a*X
Lemmas referenced :
Cauchy-Schwarz-strict,
real-vec-norm-positive-iff,
all_wf,
real_wf,
real-vec-sep_wf,
real-vec-mul_wf,
rless_wf,
int-to-real_wf,
real-vec-norm_wf,
real-vec_wf,
nat_wf,
rneq-symmetry,
rdiv_wf,
real-vec-sep-iff,
rmul_wf,
rneq-iff-rabs,
rneq_wf,
exists_wf,
int_seg_wf,
rmul_preserves_rless,
rabs_wf,
rsub_wf,
rabs-neq-zero,
radd_wf,
rminus_wf,
rmul-zero-both,
rinv_wf2,
rmul_comm,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermMinus_wf,
req-iff-rsub-is-0,
rless_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rabs_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
itermMultiply_wf,
itermConstant_wf,
req_inversion,
rabs-rmul,
radd_functionality,
rminus_functionality,
rmul-rinv,
real_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
independent_functionElimination,
because_Cache,
rename,
isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
independent_isectElimination,
dependent_pairFormation,
setElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. ((r0 < ||y||) {}\mRightarrow{} (\mforall{}a:\mBbbR{}. x \mneq{} a*y) {}\mRightarrow{} (|x\mcdot{}y| < (||x|| * ||y||)))
Date html generated:
2017_10_03-AM-11_02_44
Last ObjectModification:
2017_06_19-PM-05_19_37
Theory : reals
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