Nuprl Lemma : rv-ip-add
∀[rv:InnerProductSpace]. ∀[x,y,z:Point]. (x + y ⋅ z = (x ⋅ z + y ⋅ z))
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
rv-add: x + y,
ss-point: Point,
req: x = y,
radd: a + b,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
rv-ip: x ⋅ y,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
btrue: tt,
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
subtype_rel: A ⊆r B,
record-select: r.x,
record+: record+,
inner-product-space: InnerProductSpace,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
separation-space_wf,
real-vector-space_wf,
inner-product-space_wf,
subtype_rel_transitivity,
inner-product-space_subtype,
rv-ip_wf,
req_witness,
sq_stable__req,
exists_wf,
int-to-real_wf,
rless_wf,
rv-0_wf,
ss-sep_wf,
rmul_wf,
rv-mul_wf,
radd_wf,
rv-add_wf,
req_wf,
ss-eq_wf,
all_wf,
real_wf,
real-vector-space_subtype1,
ss-point_wf,
subtype_rel_self
Rules used in proof :
dependent_functionElimination,
isect_memberEquality,
independent_isectElimination,
instantiate,
productElimination,
independent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
Error :applyLambdaEquality,
rename,
setElimination,
natural_numberEquality,
functionExtensionality,
hypothesisEquality,
lambdaEquality,
productEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
functionEquality,
setEquality,
isectElimination,
extract_by_obid,
tokenEquality,
applyEquality,
hypothesis,
thin,
dependentIntersectionEqElimination,
sqequalRule,
dependentIntersectionElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[rv:InnerProductSpace]. \mforall{}[x,y,z:Point]. (x + y \mcdot{} z = (x \mcdot{} z + y \mcdot{} z))
Date html generated:
2016_11_08-AM-09_14_50
Last ObjectModification:
2016_10_31-PM-04_27_44
Theory : inner!product!spaces
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