Nuprl Lemma : rv-norm-mul
∀[rv:InnerProductSpace]. ∀[x:Point]. ∀[a:ℝ]. (||a*x|| = (|a| * ||x||))
Proof
Definitions occuring in Statement :
rv-norm: ||x||,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
ss-point: Point,
rabs: |x|,
req: x = y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rabs-of-nonneg,
rmul_comm,
square-nonneg,
rnexp2,
rabs_functionality,
rabs-rnexp,
req_inversion,
rmul-assoc,
rnexp-rmul,
rv-ip-mul2,
req_weakening,
rmul_functionality,
rv-ip-mul,
rv-norm-squared,
req_transitivity,
req_functionality,
uiff_transitivity,
rv-ip-symmetry,
le_wf,
false_wf,
rnexp_wf,
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
real-vector-space_subtype1,
ss-point_wf,
inner-product-space_subtype,
req_witness,
zero-rleq-rabs,
rmul-nonneg-case1,
rv-norm-nonneg,
rabs_wf,
rv-ip_wf,
rmul_wf,
req_wf,
int-to-real_wf,
rleq_wf,
real_wf,
rv-mul_wf,
rv-norm_wf,
square-req-iff
Rules used in proof :
lambdaFormation,
dependent_set_memberEquality,
instantiate,
isect_memberEquality,
productElimination,
independent_pairFormation,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
sqequalRule,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x:Point]. \mforall{}[a:\mBbbR{}]. (||a*x|| = (|a| * ||x||))
Date html generated:
2016_11_08-AM-09_16_46
Last ObjectModification:
2016_11_01-AM-00_42_41
Theory : inner!product!spaces
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