Nuprl Lemma : real-vec-angle-lemma
∀n:ℕ. ∀x,z:ℝ^n. (d(z;x) < d(z;r(-1)*x) ⇐⇒ r0 < x⋅z)
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],
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
rsqrt-rless-iff,
dot-product-nonneg,
real-vec-sub_wf,
dot-product_wf,
rleq_wf,
int-to-real_wf,
real-vec-mul_wf,
rless_wf,
rsqrt_wf,
real_wf,
req_wf,
rmul_wf,
iff_wf,
real-vec_wf,
nat_wf,
dot-product-comm,
radd_wf,
rless_functionality,
rsub_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_transitivity,
dot-product-linearity1-sub,
rsub_functionality,
req_weakening,
dot-product-linearity2,
rmul_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
radd-preserves-rless,
rmul_preserves_rless,
rless-int,
rmul-zero-both,
rmul_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
introduction,
extract_by_obid,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
because_Cache,
minusEquality,
independent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
independent_isectElimination,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,z:\mBbbR{}\^{}n. (d(z;x) < d(z;r(-1)*x) \mLeftarrow{}{}\mRightarrow{} r0 < x\mcdot{}z)
Date html generated:
2018_05_22-PM-02_25_51
Last ObjectModification:
2018_03_27-PM-10_36_41
Theory : reals
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