Nuprl Lemma : rv-mul-add
∀[rv:RealVectorSpace]. ∀[a,b:ℝ]. ∀[x:Point]. a*x + b*x ≡ a + b*x
Proof
Definitions occuring in Statement :
rv-mul: a*x,
rv-add: x + y,
real-vector-space: RealVectorSpace,
radd: a + b,
real: ℝ,
ss-eq: x ≡ y,
ss-point: Point,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
ss-eq: x ≡ y,
not: ¬A,
false: False,
prop: ℙ,
real-vector-space: RealVectorSpace,
record+: record+,
record-select: r.x,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
or: P ∨ Q,
rv-mul: a*x,
rv-add: x + y,
squash: ↓T,
sq_stable: SqStable(P)
Lemmas referenced :
ss-eq_inversion,
real-vector-space_subtype1,
rv-mul_wf,
radd_wf,
rv-add_wf,
ss-sep_wf,
ss-point_wf,
real_wf,
real-vector-space_wf,
subtype_rel_self,
all_wf,
ss-eq_wf,
or_wf,
int-to-real_wf,
rmul_wf,
rneq_wf,
sq_stable__ss-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
isectElimination,
independent_functionElimination,
lambdaEquality,
because_Cache,
isect_memberEquality,
voidElimination,
dependentIntersectionElimination,
dependentIntersectionEqElimination,
tokenEquality,
setEquality,
functionEquality,
productEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination
Latex:
\mforall{}[rv:RealVectorSpace]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[x:Point]. a*x + b*x \mequiv{} a + b*x
Date html generated:
2017_10_04-PM-11_50_23
Last ObjectModification:
2017_06_22-PM-06_44_29
Theory : inner!product!spaces
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