Nuprl Lemma : rv-orthogonal-isometry
∀[rv:InnerProductSpace]. ∀[f:Point ⟶ Point]. Isometry(f) supposing Orthogonal(f)
Proof
Definitions occuring in Statement :
rv-isometry: Isometry(f),
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rv-minus: -x,
rv-sub: x - y,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
rv-isometry: Isometry(f),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-add_functionality,
ss-eq_transitivity,
ss-eq_functionality,
ss-eq_weakening,
rv-mul_wf,
rv-add_wf,
req_weakening,
rv-norm_functionality,
req_functionality,
rv-orthogonal_wf,
rmul_wf,
int-to-real_wf,
rleq_wf,
real_wf,
rv-ip_wf,
req_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
real-vector-space_subtype1,
inner-product-space_subtype,
rv-sub_wf,
rv-norm_wf,
ss-point_wf,
req_witness,
rv-orthogonal-iff-norm-preserving
Rules used in proof :
minusEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
independent_isectElimination,
instantiate,
functionExtensionality,
isect_memberEquality,
sqequalRule,
applyEquality,
because_Cache,
independent_functionElimination,
productElimination,
dependent_functionElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[f:Point {}\mrightarrow{} Point]. Isometry(f) supposing Orthogonal(f)
Date html generated:
2016_11_08-AM-09_18_21
Last ObjectModification:
2016_11_02-PM-08_44_49
Theory : inner!product!spaces
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