Nuprl Lemma : Cauchy-Schwarz-non-equality
∀n:ℕ. ∀x,y:ℝ^n. ((r0 < ||y||) ⇒ x ≠ (||x||/||y||)*y ⇒ x ≠ (-(||x||)/||y||)*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,
rdiv: (x/y),
rless: x < y,
rabs: |x|,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
real-vec-sep: a ≠ b,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
not: ¬A,
top: Top
Lemmas referenced :
Cauchy-Schwarz-non-equality1,
real_wf,
real-vec-sep_wf,
real-vec-mul_wf,
rdiv_wf,
rminus_wf,
real-vec-norm_wf,
rless_wf,
int-to-real_wf,
real-vec_wf,
nat_wf,
real-vec-sep-cases,
real-vec-dist_wf,
rmul_wf,
rabs_wf,
rsub_wf,
rmul-is-positive,
rneq-iff-rabs,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
rless_functionality,
req_weakening,
real-vec-dist-vec-mul,
rneq_functionality,
real-vec-norm-mul,
rmul_preserves_req,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
req-iff-rsub-is-0,
itermConstant_wf,
req_functionality,
req_transitivity,
rminus_functionality,
rmul_functionality,
rmul-rinv,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rneq-rabs,
rmul_preserves_rneq_iff2,
rmul-one,
rneq-symmetry,
equal_wf,
length-rneq-real-vec-sep,
rneq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
because_Cache,
independent_isectElimination,
sqequalRule,
inrFormation,
natural_numberEquality,
unionElimination,
applyEquality,
productElimination,
inlFormation,
voidElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
promote_hyp,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n.
((r0 < ||y||) {}\mRightarrow{} x \mneq{} (||x||/||y||)*y {}\mRightarrow{} x \mneq{} (-(||x||)/||y||)*y {}\mRightarrow{} (|x\mcdot{}y| < (||x|| * ||y||)))
Date html generated:
2017_10_03-AM-11_03_11
Last ObjectModification:
2017_06_20-AM-11_40_47
Theory : reals
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