Nuprl Lemma : rv-pos-angle-lemma
∀n:ℕ. ∀x,y:ℝ^n. ((||x|| = ||y||) ⇒ (r0 < d(x;y)) ⇒ (r0 < d(r(-1)*x;y)) ⇒ (|x⋅y| < (||x|| * ||y||)))
Proof
Definitions occuring in Statement :
real-vec-dist: d(x;y),
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-mul: a*X,
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
req: x = y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
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],
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
real-vec-dist: d(x;y),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
or: P ∨ Q,
rsub: x - y,
cand: A c∧ B,
nat_plus: ℕ+,
true: True
Lemmas referenced :
square-rless-implies,
rabs_wf,
dot-product_wf,
rmul_wf,
real-vec-norm_wf,
rmul-nonneg-case1,
real-vec-norm-nonneg,
rless_wf,
int-to-real_wf,
real-vec-dist_wf,
real-vec-mul_wf,
real_wf,
rleq_wf,
req_wf,
real-vec_wf,
nat_wf,
rnexp_wf,
false_wf,
le_wf,
rnexp2-nonneg,
rless_functionality,
req_inversion,
rabs-rnexp,
rnexp-rmul,
rabs-of-nonneg,
rmul_functionality,
real-vec-norm-squared,
rnexp2,
req_weakening,
rnexp-positive,
real-vec-sub_wf,
radd_wf,
real-vec-norm-diff-squared,
radd_functionality,
dot-product-linearity2,
req_transitivity,
rmul-assoc,
rmul-int,
rmul-one-both,
radd-assoc,
radd_comm,
req_functionality,
rnexp_functionality,
radd-preserves-req,
rsub_wf,
rminus_wf,
rmul-is-positive,
rless-int,
less_than_wf,
or_wf,
uiff_transitivity,
rmul-identity1,
rmul-distrib2,
radd-int,
rmul-distrib,
rmul_over_rminus,
radd-ac,
radd-rminus-both,
radd-zero-both,
rmul-zero-both,
radd-preserves-rless,
rabs-rless-iff,
rminus-as-rmul,
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,
independent_functionElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
natural_numberEquality,
minusEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
sqequalRule,
dependent_set_memberEquality,
productElimination,
multiplyEquality,
promote_hyp,
addEquality,
addLevel,
orFunctionality,
andLevelFunctionality,
imageElimination,
voidElimination,
productEquality,
unionElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n.
((||x|| = ||y||) {}\mRightarrow{} (r0 < d(x;y)) {}\mRightarrow{} (r0 < d(r(-1)*x;y)) {}\mRightarrow{} (|x\mcdot{}y| < (||x|| * ||y||)))
Date html generated:
2017_10_03-AM-11_06_12
Last ObjectModification:
2017_03_02-PM-04_07_58
Theory : reals
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