Nuprl Lemma : rv-ip-mul2
∀[rv:InnerProductSpace]. ∀[a:ℝ]. ∀[x,y:Point(rv)]. (y ⋅ a*x = (a * y ⋅ x))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-mul: a*x,
req: x = y,
rmul: a * b,
real: ℝ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rv-ip_wf,
rv-mul_wf,
inner-product-space_subtype,
rmul_wf,
Error :ss-point_wf,
real-vector-space_subtype1,
subtype_rel_transitivity,
inner-product-space_wf,
real-vector-space_wf,
Error :separation-space_wf,
real_wf,
rv-ip-mul,
req_functionality,
rv-ip-symmetry,
rmul_functionality,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
independent_functionElimination,
inhabitedIsType,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
universeIsType,
instantiate,
independent_isectElimination,
productElimination
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[a:\mBbbR{}]. \mforall{}[x,y:Point(rv)]. (y \mcdot{} a*x = (a * y \mcdot{} x))
Date html generated:
2020_05_20-PM-01_11_02
Last ObjectModification:
2019_12_09-PM-11_53_21
Theory : inner!product!spaces
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