Nuprl Lemma : rv-ip_wf
∀[rv:InnerProductSpace]. ∀[x,y:Point]. (x ⋅ y ∈ ℝ)
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
ss-point: Point,
real: ℝ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
record-select: r.x,
record+: record+,
inner-product-space: InnerProductSpace,
rv-ip: x ⋅ y,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
int-to-real_wf,
rless_wf,
rv-0_wf,
ss-sep_wf,
rmul_wf,
rv-mul_wf,
radd_wf,
rv-add_wf,
req_wf,
all_wf,
real_wf,
real-vector-space_subtype1,
ss-point_wf,
subtype_rel_self,
inner-product-space_subtype
Rules used in proof :
isect_memberEquality,
independent_isectElimination,
instantiate,
axiomEquality,
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
lambdaEquality,
productEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
functionEquality,
setEquality,
isectElimination,
tokenEquality,
thin,
dependentIntersectionEqElimination,
dependentIntersectionElimination,
sqequalRule,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
applyEquality,
hypothesisEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y:Point]. (x \mcdot{} y \mmember{} \mBbbR{})
Date html generated:
2016_11_08-AM-09_14_41
Last ObjectModification:
2016_10_31-PM-02_34_37
Theory : inner!product!spaces
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