Nuprl Lemma : trans-from-kernel_functionality
∀rv:InnerProductSpace. ∀e:{e:Point| e^2 = r1} . ∀f,g:{h:Point| h ⋅ e = r0} ⟶ ℝ ⟶ ℝ.
(trans-kernel-fun(rv;e;f)
⇒ (∀h:{h:Point| h ⋅ e = r0} . ∀r:ℝ. ((f h (g h r)) = r))
⇒ (∀s,t:ℝ. ∀x,y:Point. ((s = t) ⇒ x ≡ y ⇒ trans-from-kernel(rv;e;f;g;s;x) ≡ trans-from-kernel(rv;e;f;g;t;y))))
Proof
Definitions occuring in Statement :
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
trans-kernel-fun: trans-kernel-fun(rv;e;f),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
ss-eq: x ≡ y,
not: ¬A,
or: P ∨ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
trans-from-kernel-sep,
ss-sep_wf,
real-vector-space_subtype1,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
trans-from-kernel_wf,
ss-point_wf,
req_wf,
rv-ip_wf,
int-to-real_wf,
ss-eq_wf,
real_wf,
all_wf,
trans-kernel-fun_wf,
set_wf,
rneq_irreflexivity,
rneq_functionality,
req_weakening
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
unionElimination,
isectElimination,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
functionExtensionality,
setEquality,
natural_numberEquality,
lambdaEquality,
functionEquality,
voidElimination,
productElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}e:\{e:Point| e\^{}2 = r1\} . \mforall{}f,g:\{h:Point| h \mcdot{} e = r0\} {}\mrightarrow{} \mBbbR{} {}\mrightarrow{} \mBbbR{}.
(trans-kernel-fun(rv;e;f)
{}\mRightarrow{} (\mforall{}h:\{h:Point| h \mcdot{} e = r0\} . \mforall{}r:\mBbbR{}. ((f h (g h r)) = r))
{}\mRightarrow{} (\mforall{}s,t:\mBbbR{}. \mforall{}x,y:Point.
((s = t) {}\mRightarrow{} x \mequiv{} y {}\mRightarrow{} trans-from-kernel(rv;e;f;g;s;x) \mequiv{} trans-from-kernel(rv;e;f;g;t;y))))
Date html generated:
2017_10_05-AM-00_24_32
Last ObjectModification:
2017_06_30-PM-02_06_32
Theory : inner!product!spaces
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