Nuprl Lemma : trans-from-kernel-sep
∀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. (trans-from-kernel(rv;e;f;g;s;x) # trans-from-kernel(rv;e;f;g;t;y) ⇒ (x # y ∨ s ≠ t))))
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,
rneq: x ≠ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-sep: x # y,
ss-point: Point,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: 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],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
and: P ∧ Q,
exists: ∃x:A. B[x],
trans-kernel-fun: trans-kernel-fun(rv;e;f),
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
trans-from-kernel: trans-from-kernel(rv;e;f;g;t;x),
rv-decomp: rv-decomp(rv;x;e),
or: P ∨ Q,
true: True
Lemmas referenced :
kernel-fun-properties,
sq_stable__req,
rv-ip_wf,
int-to-real_wf,
real_wf,
all_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,
trans-kernel-fun_wf,
set_wf,
rv-sub_wf,
rv-mul_wf,
rsub_wf,
rmul_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
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,
rv-sub-sep,
ss-sep_wf,
rv-add-sep,
radd_wf,
rneq_wf,
ss-sep-irrefl,
rv-add_wf,
equal_wf,
rv-mul-sep,
rv-ip-rneq,
member_wf,
squash_wf,
true_wf,
rneq-radd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
isectElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
promote_hyp,
setEquality,
applyEquality,
instantiate,
independent_isectElimination,
lambdaEquality,
functionExtensionality,
dependent_set_memberEquality,
functionEquality,
approximateComputation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
inlFormation,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeEquality,
applyLambdaEquality,
inrFormation
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.
(trans-from-kernel(rv;e;f;g;s;x) \# trans-from-kernel(rv;e;f;g;t;y) {}\mRightarrow{} (x \# y \mvee{} s \mneq{} t))))
Date html generated:
2017_10_05-AM-00_24_27
Last ObjectModification:
2017_06_30-PM-02_05_58
Theory : inner!product!spaces
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