Nuprl Lemma : rv-norm-squared
∀[rv:InnerProductSpace]. ∀[x:Point]. (||x||^2 = x^2)
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
ss-point: Point,
rnexp: x^k1,
req: x = y,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
guard: {T},
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
req_weakening,
rnexp2,
req_functionality,
sq_stable__req,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
real-vector-space_subtype1,
ss-point_wf,
le_wf,
false_wf,
rnexp_wf,
req_witness,
equal_wf,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rleq_wf,
real_wf,
set_wf,
rv-norm_wf
Rules used in proof :
imageElimination,
baseClosed,
imageMemberEquality,
productElimination,
because_Cache,
isect_memberEquality,
independent_isectElimination,
instantiate,
setEquality,
rename,
setElimination,
applyEquality,
independent_pairFormation,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
natural_numberEquality,
productEquality,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. (||x||\^{}2 = x\^{}2)
Date html generated:
2016_11_08-AM-09_16_19
Last ObjectModification:
2016_10_31-PM-04_39_29
Theory : inner!product!spaces
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