Nuprl Lemma : rv-pos-angle-permute-lemma
∀n:ℕ. ∀x,y:ℝ^n. ((|x⋅y| < (||x|| * ||y||)) ⇒ (|x⋅y - x| < (||x|| * ||y - x||)))
Proof
Definitions occuring in Statement :
real-vec-norm: ||x||,
dot-product: x⋅y,
real-vec-sub: X - Y,
real-vec: ℝ^n,
rless: x < y,
rabs: |x|,
rmul: a * b,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q
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,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rsub: x - y
Lemmas referenced :
square-rless-implies,
rabs_wf,
dot-product_wf,
real-vec-sub_wf,
rmul_wf,
real-vec-norm_wf,
rmul-nonneg-case1,
real-vec-norm-nonneg,
rnexp-rless,
zero-rleq-rabs,
less_than_wf,
rless_wf,
real-vec_wf,
nat_wf,
rnexp_wf,
false_wf,
le_wf,
rnexp2-nonneg,
rless_functionality,
req_inversion,
rabs-rnexp,
req_transitivity,
rnexp-rmul,
rmul_functionality,
real-vec-norm-squared,
rabs-of-nonneg,
req_weakening,
rnexp2,
rsub_wf,
dot-product-comm,
radd_wf,
int-to-real_wf,
rminus_wf,
dot-product-linearity1-sub,
rsub_functionality,
radd-ac,
radd_functionality,
radd_comm,
rminus-rminus,
radd-int,
rminus-as-rmul,
radd-assoc,
rmul-distrib2,
rminus-radd,
rminus_functionality,
rmul_comm,
rmul-distrib,
rmul_over_rminus,
radd-preserves-rless,
radd-zero-both,
rmul-zero-both,
rmul-identity1
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,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
productElimination,
minusEquality,
addEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. ((|x\mcdot{}y| < (||x|| * ||y||)) {}\mRightarrow{} (|x\mcdot{}y - x| < (||x|| * ||y - x||)))
Date html generated:
2017_10_03-AM-10_57_36
Last ObjectModification:
2017_03_02-AM-10_57_59
Theory : reals
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