Nuprl Lemma : rv-orthogonal-implies-functional
∀[rv:InnerProductSpace]. ∀f:Point ⟶ Point. (Orthogonal(f) ⇒ (∀x,y:Point. (x ≡ y ⇒ f x ≡ f y)))
Proof
Definitions occuring in Statement :
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rv-orthogonal-isometry,
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,
rv-orthogonal_wf,
rv-isometry-implies-functional
Rules used in proof :
because_Cache,
functionEquality,
sqequalRule,
independent_isectElimination,
instantiate,
applyEquality,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}f:Point {}\mrightarrow{} Point. (Orthogonal(f) {}\mRightarrow{} (\mforall{}x,y:Point. (x \mequiv{} y {}\mRightarrow{} f x \mequiv{} f y)))
Date html generated:
2016_11_08-AM-09_18_47
Last ObjectModification:
2016_11_02-PM-08_50_01
Theory : inner!product!spaces
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