Nuprl Lemma : rv-orthog-ext_wf
∀rv:InnerProductSpace. ∀f:Point ⟶ Point.
rv-orthog-ext(rv;f) ∈ ∀x,y:Point. (f x # f y ⇒ x # y) supposing Orthogonal(f)
Proof
Definitions occuring in Statement :
rv-orthog-ext: rv-orthog-ext(rv;f),
rv-orthogonal: Orthogonal(f),
inner-product-space: InnerProductSpace,
ss-sep: x # y,
ss-point: Point,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rv-orthog-ext: rv-orthog-ext(rv;f),
rv-orthogonal-implies-extensional-ext,
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
ss-sep_wf,
rv-orthogonal_wf,
isect_wf,
separation-space_wf,
real-vector-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-point_wf,
inner-product-space_wf,
all_wf,
rv-orthogonal-implies-extensional-ext
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
universeEquality,
cumulativity,
functionExtensionality,
dependent_functionElimination,
because_Cache,
independent_isectElimination,
functionEquality,
isectElimination,
hypothesisEquality,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
introduction,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}f:Point {}\mrightarrow{} Point.
rv-orthog-ext(rv;f) \mmember{} \mforall{}x,y:Point. (f x \# f y {}\mRightarrow{} x \# y) supposing Orthogonal(f)
Date html generated:
2016_11_08-AM-09_18_40
Last ObjectModification:
2016_11_02-PM-04_27_39
Theory : inner!product!spaces
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