Nuprl Lemma : rv-perp-1
∀rv:InnerProductSpace. ∀x:Point(rv). (x # 0 ⇒ (∃y:Point(rv). ((y^2 = r1) ∧ (x ⋅ y = r0))))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
req: x = y,
int-to-real: r(n),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
inner-product-space: InnerProductSpace,
record+: record+,
record-select: r.x,
subtype_rel: A ⊆r B,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
rv-ip: x ⋅ y,
uimplies: b supposing a,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
subtype_rel_self,
Error :ss-point_wf,
real-vector-space_subtype1,
real_wf,
all_wf,
Error :ss-eq_wf,
req_wf,
rv-add_wf,
radd_wf,
rv-mul_wf,
rmul_wf,
iff_wf,
Error :ss-sep_wf,
rv-0_wf,
rless_wf,
int-to-real_wf,
exists_wf,
inner-product-space_subtype,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-unit_wf,
rv-unit-squared,
rv-ip_wf,
rv-unit-property,
req_functionality,
rv-ip_functionality,
Error :ss-eq_weakening,
req_weakening,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
req-iff-rsub-is-0,
req_transitivity,
rv-ip-mul2,
rmul_functionality,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
sqequalHypSubstitution,
dependentIntersectionElimination,
sqequalRule,
dependentIntersectionEqElimination,
thin,
hypothesis,
applyEquality,
tokenEquality,
extract_by_obid,
isectElimination,
setEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productEquality,
lambdaEquality_alt,
hypothesisEquality,
inhabitedIsType,
universeIsType,
closedConclusion,
natural_numberEquality,
applyLambdaEquality,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination,
productElimination,
instantiate,
independent_isectElimination,
dependent_pairFormation_alt,
independent_pairFormation,
productIsType,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}rv:InnerProductSpace. \mforall{}x:Point(rv). (x \# 0 {}\mRightarrow{} (\mexists{}y:Point(rv). ((y\^{}2 = r1) \mwedge{} (x \mcdot{} y = r0))))
Date html generated:
2020_05_20-PM-01_11_40
Last ObjectModification:
2019_12_09-PM-07_24_32
Theory : inner!product!spaces
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