Nuprl Lemma : not-rv-pos-angle
∀n:ℕ. ∀a,b,c:ℝ^n.
((r0 < d(a;b)) ⇒ (r0 < d(c;b)) ⇒ (¬rv-pos-angle(n;a;b;c)) ⇒ (∃t:ℝ. ((r0 < |t|) ∧ req-vec(n;c;b + t*a - b))))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
rv-pos-angle: rv-pos-angle(n;a;b;c),
real-vec-mul: a*X,
real-vec-sub: X - Y,
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
real-vec-dist: d(x;y),
rv-pos-angle: rv-pos-angle(n;a;b;c),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
rdiv: (x/y),
cand: A c∧ B,
exists: ∃x:A. B[x],
req-vec: req-vec(n;x;y),
real-vec-add: X + Y,
real-vec-sub: X - Y,
rsub: x - y,
real-vec: ℝ^n
Lemmas referenced :
not-rless,
rabs_wf,
dot-product_wf,
real-vec-sub_wf,
rmul_wf,
real-vec-norm_wf,
Cauchy-Schwarz,
rleq_antisymmetry,
req_functionality,
rabs_functionality,
dot-product-comm,
rmul_comm,
not_wf,
rv-pos-angle_wf,
rless_wf,
int-to-real_wf,
real-vec-dist_wf,
real_wf,
rleq_wf,
real-vec_wf,
nat_wf,
rnexp-rless,
rleq_weakening_equal,
less_than_wf,
rnexp_wf,
false_wf,
le_wf,
rless_functionality,
rnexp0,
req_weakening,
rnexp2-nonneg,
rdiv_wf,
rmul_preserves_rless,
rinv_wf2,
rabs-of-nonneg,
rabs-rdiv,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_transitivity,
rmul_functionality,
rmul-rinv,
rmul-is-positive,
real-vec-mul_wf,
real-vec-add_wf,
req-vec_wf,
equal_wf,
int_seg_wf,
radd-rminus-assoc,
radd_comm,
radd_functionality,
uiff_transitivity,
rminus_wf,
radd_wf,
req_wf,
rsub_wf,
radd-preserves-req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
because_Cache,
independent_functionElimination,
productElimination,
natural_numberEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
dependent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
inrFormation,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
inlFormation,
productEquality,
dependent_pairFormation,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a,b,c:\mBbbR{}\^{}n.
((r0 < d(a;b))
{}\mRightarrow{} (r0 < d(c;b))
{}\mRightarrow{} (\mneg{}rv-pos-angle(n;a;b;c))
{}\mRightarrow{} (\mexists{}t:\mBbbR{}. ((r0 < |t|) \mwedge{} req-vec(n;c;b + t*a - b))))
Date html generated:
2017_10_03-AM-10_58_48
Last ObjectModification:
2017_07_28-AM-08_22_00
Theory : reals
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