Nuprl Lemma : rv-orthogonal-iff-norm-preserving
∀[rv:InnerProductSpace]
∀f:Point ⟶ Point
(Orthogonal(f) ⇐⇒ (∀x,y:Point. f x + y ≡ f x + f y) ∧ (∀x:Point. ((∀a:ℝ. f a*x ≡ a*f x) ∧ (||f x|| = ||x||))))
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
ss-eq: x ≡ y,
ss-point: Point,
req: x = y,
real: ℝ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
true: True,
squash: ↓T,
less_than: a < b,
or: P ∨ Q,
rneq: x ≠ y,
rsub: x - y,
rv-minus: -x,
rv-sub: x - y,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rv-norm: ||x||,
rv-orthogonal: Orthogonal(f),
false: False,
not: ¬A,
ss-eq: x ≡ y,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmul_comm,
rless_wf,
rless-int,
rmul_preserves_req,
radd-zero-both,
rmul-zero-both,
radd-int,
rmul_functionality,
rminus-as-rmul,
rmul-distrib2,
rmul-identity1,
radd-rminus-assoc,
radd_comm,
radd-ac,
radd-assoc,
rminus-radd,
rminus_wf,
radd-preserves-req,
rsub_functionality,
radd_functionality,
rv-ip-sub-squared,
rsub_wf,
radd_wf,
rv-sub_wf,
ss-eq_inversion,
ss-eq_weakening,
rv-add_functionality,
rv-ip_functionality,
uiff_transitivity,
rnexp_functionality,
req_transitivity,
rv-norm-squared,
le_wf,
false_wf,
rnexp_wf,
req_inversion,
rsqrt_functionality,
req_functionality,
req_weakening,
rv-ip-nonneg,
rsqrt_wf,
req_witness,
ss-sep_wf,
rv-ip_wf,
rmul_wf,
int-to-real_wf,
rleq_wf,
rv-norm_wf,
req_wf,
rv-mul_wf,
rv-add_wf,
ss-eq_wf,
all_wf,
rv-orthogonal_wf,
real_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-point_wf
Rules used in proof :
baseClosed,
imageMemberEquality,
inrFormation,
addEquality,
promote_hyp,
allFunctionality,
minusEquality,
dependent_set_memberEquality,
independent_functionElimination,
voidElimination,
independent_pairEquality,
functionEquality,
natural_numberEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
productEquality,
productElimination,
functionExtensionality,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
instantiate,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]
\mforall{}f:Point {}\mrightarrow{} Point
(Orthogonal(f)
\mLeftarrow{}{}\mRightarrow{} (\mforall{}x,y:Point. f x + y \mequiv{} f x + f y)
\mwedge{} (\mforall{}x:Point. ((\mforall{}a:\mBbbR{}. f a*x \mequiv{} a*f x) \mwedge{} (||f x|| = ||x||))))
Date html generated:
2016_11_08-AM-09_18_07
Last ObjectModification:
2016_11_01-AM-00_25_56
Theory : inner!product!spaces
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