Nuprl Lemma : ip-triangle-permute-lemma
∀rv:InnerProductSpace. ∀x,y:Point(rv). ((|x ⋅ y| < (||x|| * ||y||)) ⇒ (|x ⋅ y - x| < (||x|| * ||y - x||)))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-sub: x - y,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rless: x < y,
rabs: |x|,
rmul: a * b,
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],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
prop: ℙ,
false: False,
guard: {T},
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
square-rless-implies,
rabs_wf,
rv-ip_wf,
rv-sub_wf,
inner-product-space_subtype,
rmul_wf,
rv-norm_wf,
rmul-nonneg-case1,
rv-norm-nonneg,
rnexp-rless,
zero-rleq-rabs,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rless_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rnexp_wf,
istype-void,
istype-le,
rnexp2-nonneg,
rless_functionality,
req_inversion,
rabs-rnexp,
req_transitivity,
rnexp-rmul,
rmul_functionality,
rv-norm-squared,
rabs-of-nonneg,
req_weakening,
rnexp2,
rsub_wf,
radd_wf,
int-to-real_wf,
rv-ip-sub2,
rv-ip-sub-squared,
rv-ip-symmetry,
radd_functionality,
rsub_functionality,
radd-preserves-rless,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
independent_functionElimination,
independent_isectElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
Error :memTop,
universeIsType,
voidElimination,
instantiate,
productElimination,
int_eqEquality,
isect_memberEquality_alt
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x,y:Point(rv).
((|x \mcdot{} y| < (||x|| * ||y||)) {}\mRightarrow{} (|x \mcdot{} y - x| < (||x|| * ||y - x||)))
Date html generated:
2020_05_20-PM-01_13_19
Last ObjectModification:
2019_12_10-AM-00_46_40
Theory : inner!product!spaces
Home
Index