Nuprl Lemma : rv-decomp_wf
∀[rv:InnerProductSpace]. ∀[e,x:Point]. rv-decomp(rv;x;e) ∈ {p:Point × ℝ| x ≡ fst(p) + snd(p)*e ∧ (fst(p) ⋅ e = r0)} su\000Cpposing e^2 = r1
Proof
Definitions occuring in Statement :
rv-decomp: rv-decomp(rv;x;e),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
rv-add: x + y,
req: x = y,
int-to-real: r(n),
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
product: x:A × B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rv-decomp: rv-decomp(rv;x;e),
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
rv-sub_wf,
inner-product-space_subtype,
rv-mul_wf,
rv-ip_wf,
ss-eq_wf,
rv-add_wf,
req_wf,
int-to-real_wf,
ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
separation-space_wf,
ss-eq_weakening,
rsub_wf,
rmul_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
req-iff-rsub-is-0,
ss-eq_functionality,
rv-sub-add,
req_functionality,
rv-ip-sub,
req_weakening,
rsub_functionality,
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,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
independent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
independent_pairFormation,
productEquality,
productElimination,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
instantiate,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
approximateComputation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[e,x:Point].
rv-decomp(rv;x;e) \mmember{} \{p:Point \mtimes{} \mBbbR{}| x \mequiv{} fst(p) + snd(p)*e \mwedge{} (fst(p) \mcdot{} e = r0)\} supposing e\^{}2 = r1
Date html generated:
2017_10_05-AM-00_20_59
Last ObjectModification:
2017_06_21-PM-02_28_55
Theory : inner!product!spaces
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