Nuprl Lemma : ip-triangle-lemma
∀rv:InnerProductSpace. ∀x,y:Point.
((||x|| = ||y||) ⇒ (r0 < ||x - y||) ⇒ (r0 < ||r(-1)*x - y||) ⇒ (|x ⋅ y| < (||x|| * ||y||)))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rless: x < y,
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
or: P ∨ Q,
cand: A c∧ B,
nat_plus: ℕ+,
true: True
Lemmas referenced :
square-rless-implies,
rabs_wf,
rv-ip_wf,
rmul_wf,
rv-norm_wf,
real_wf,
rleq_wf,
int-to-real_wf,
req_wf,
rmul-nonneg-case1,
rv-norm-nonneg,
rless_wf,
rv-sub_wf,
inner-product-space_subtype,
rv-mul_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
rnexp_wf,
false_wf,
le_wf,
rnexp2-nonneg,
rnexp-positive,
radd_wf,
rsub_wf,
rless_functionality,
req_inversion,
rabs-rnexp,
rnexp-rmul,
rabs-of-nonneg,
rmul_functionality,
rv-norm-squared,
rnexp2,
req_weakening,
req_transitivity,
rv-ip-sub-squared,
radd_functionality,
rsub_functionality,
rv-ip-mul,
rv-ip-mul2,
rmul-assoc,
rmul-int,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_functionality,
rnexp_functionality,
radd-preserves-req,
rmul-is-positive,
rless-int,
less_than_wf,
or_wf,
radd-preserves-rless,
rminus_wf,
rabs-rless-iff,
itermMinus_wf,
real_term_value_minus_lemma,
rnexp-rless,
zero-rleq-rabs
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
productEquality,
natural_numberEquality,
sqequalRule,
independent_functionElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
minusEquality,
instantiate,
dependent_set_memberEquality,
productElimination,
multiplyEquality,
promote_hyp,
computeAll,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addLevel,
orFunctionality,
andLevelFunctionality,
imageElimination,
unionElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x,y:Point.
((||x|| = ||y||) {}\mRightarrow{} (r0 < ||x - y||) {}\mRightarrow{} (r0 < ||r(-1)*x - y||) {}\mRightarrow{} (|x \mcdot{} y| < (||x|| * ||y||)))
Date html generated:
2017_10_04-PM-11_59_17
Last ObjectModification:
2017_07_28-AM-08_54_41
Theory : inner!product!spaces
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