Nuprl Lemma : rv-unit_wf
∀[rv:InnerProductSpace]. ∀[x:Point(rv)]. rv-unit(rv;x) ∈ {z:Point(rv)| z^2 = r1} supposing x # 0
Proof
Definitions occuring in Statement :
rv-unit: rv-unit(rv;x),
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-0: 0,
req: x = y,
int-to-real: r(n),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rv-unit: rv-unit(rv;x),
subtype_rel: A ⊆r B,
rneq: x ≠ y,
or: P ∨ Q,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
false: False,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
top: Top
Lemmas referenced :
rv-norm-positive,
rv-mul_wf,
inner-product-space_subtype,
rdiv_wf,
int-to-real_wf,
rv-norm_wf,
rless_wf,
req_wf,
rv-ip_wf,
Error :ss-sep_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
rv-0_wf,
Error :ss-point_wf,
rmul-is-positive,
rmul_wf,
uiff_transitivity,
req_functionality,
req_transitivity,
rv-ip-mul,
rmul_functionality,
req_weakening,
rv-ip-mul2,
rmul-assoc,
rmul-rdiv,
rnexp_wf,
istype-void,
istype-le,
rnexp2,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
req_inversion,
rv-norm-squared,
rinv-of-rmul,
rmul-rinv3,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
istype-int,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_set_memberEquality_alt,
isectElimination,
applyEquality,
sqequalRule,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
inrFormation_alt,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
instantiate,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
productElimination,
inlFormation_alt,
independent_pairFormation,
productIsType,
lambdaFormation_alt,
voidElimination,
lambdaEquality_alt,
setElimination,
rename,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point(rv)]. rv-unit(rv;x) \mmember{} \{z:Point(rv)| z\^{}2 = r1\} supposing x \# 0
Date html generated:
2020_05_20-PM-01_11_36
Last ObjectModification:
2019_12_10-AM-00_01_42
Theory : inner!product!spaces
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