Nuprl Lemma : trans-kernel_wf
∀[rv:InnerProductSpace]. ∀[e:Point]. ∀[T:ℝ ⟶ Point ⟶ Point]. ∀[t:ℝ]. ∀[h:{h:Point| h ⋅ e = r0} ]. (ρ(h;t) ∈ ℝ)
Proof
Definitions occuring in Statement :
trans-kernel: ρ(h;t),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-point: Point,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
trans-kernel: ρ(h;t),
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
rv-ip_wf,
trans-apply_wf,
real_wf,
set_wf,
ss-point_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
req_wf,
int-to-real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
functionExtensionality,
applyEquality,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
because_Cache,
isect_memberEquality,
functionEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e:Point]. \mforall{}[T:\mBbbR{} {}\mrightarrow{} Point {}\mrightarrow{} Point]. \mforall{}[t:\mBbbR{}]. \mforall{}[h:\{h:Point| h \mcdot{} e = r0\} ].
(\mrho{}(h;t) \mmember{} \mBbbR{})
Date html generated:
2017_10_05-AM-00_22_27
Last ObjectModification:
2017_06_26-PM-00_51_00
Theory : inner!product!spaces
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