Nuprl Lemma : trans-from-kernel_wf
∀[rv:InnerProductSpace]. ∀[e:{e:Point| e^2 = r1} ]. ∀[f,g:{h:Point| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ]. ∀[t:ℝ]. ∀[x:Point].
(trans-from-kernel(rv;e;f;g;t;x) ∈ Point)
Proof
Definitions occuring in Statement :
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
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 :
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
rv-decomp: rv-decomp(rv;x;e),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
req_int_terms: t1 ≡ t2,
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top
Lemmas referenced :
rv-sub_wf,
rv-mul_wf,
rv-ip_wf,
req_wf,
int-to-real_wf,
rsub_wf,
rmul_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
rv-add_wf,
inner-product-space_subtype,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
radd_wf,
real_wf,
set_wf,
req_functionality,
req_transitivity,
rv-ip-sub,
rsub_functionality,
req_weakening,
rv-ip-mul,
rmul_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
natural_numberEquality,
productElimination,
independent_isectElimination,
isect_memberFormation,
setElimination,
rename,
functionExtensionality,
setEquality,
instantiate,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
lambdaEquality,
dependent_functionElimination,
approximateComputation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e:\{e:Point| e\^{}2 = r1\} ]. \mforall{}[f,g:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}]. \mforall{}[t:\mBbbR{}].
\mforall{}[x:Point].
(trans-from-kernel(rv;e;f;g;t;x) \mmember{} Point)
Date html generated:
2017_10_05-AM-00_24_16
Last ObjectModification:
2017_06_30-PM-02_04_38
Theory : inner!product!spaces
Home
Index