Nuprl Lemma : real-vec-angle-lemma2
∀n:ℕ. ∀x,z:ℝ^n. (d(z;r(-1)*x) < d(z;x) ⇐⇒ x⋅z < r0)
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
dot-product: x⋅y,
real-vec-mul: a*X,
real-vec: ℝ^n,
rless: x < y,
int-to-real: r(n),
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
req-vec: req-vec(n;x;y),
real-vec-mul: a*X,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat: ℕ,
real-vec: ℝ^n,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
rev_implies: P ⇐ Q
Lemmas referenced :
real-vec-angle-lemma,
real-vec-mul_wf,
int-to-real_wf,
real-vec_wf,
nat_wf,
int_seg_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
real-vec-dist_wf,
real_wf,
rleq_wf,
rless_wf,
dot-product_wf,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rless_functionality,
req_weakening,
real-vec-dist_functionality,
req-vec_weakening,
iff_wf,
rless-implies-rless,
rsub_wf,
dot-product-linearity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
minusEquality,
natural_numberEquality,
hypothesis,
because_Cache,
sqequalRule,
productElimination,
setElimination,
rename,
applyEquality,
independent_isectElimination,
lambdaEquality,
setEquality,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
independent_functionElimination,
promote_hyp,
addLevel,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,z:\mBbbR{}\^{}n. (d(z;r(-1)*x) < d(z;x) \mLeftarrow{}{}\mRightarrow{} x\mcdot{}z < r0)
Date html generated:
2018_05_22-PM-02_26_09
Last ObjectModification:
2018_03_27-PM-10_53_30
Theory : reals
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