Nuprl Lemma : rv-ip_functionality
∀[rv:InnerProductSpace]. ∀[x1,x2,y1,y2:Point]. (x1 ⋅ y1 = x2 ⋅ y2) supposing (y1 ≡ y2 and x1 ≡ x2)
Proof
Definitions occuring in Statement :
rv-ip: x ⋅ y,
inner-product-space: InnerProductSpace,
ss-eq: x ≡ y,
ss-point: Point,
req: x = y,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
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,
uimplies: b supposing a,
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{}[x1,x2,y1,y2:Point]. (x1 \mcdot{} y1 = x2 \mcdot{} y2) supposing (y1 \mequiv{} y2 and x1 \mequiv{} x2)
Date html generated:
2016_11_08-AM-09_14_44
Last ObjectModification:
2016_11_02-PM-00_08_55
Theory : inner!product!spaces
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