Nuprl Lemma : formal-sum-mul-add
∀[S:Type]. ∀[K:CRng]. ∀[k,b:|K|]. ∀[x:formal-sum(K;S)]. (k +K b * x = k * x + b * x ∈ formal-sum(K;S))
Proof
Definitions occuring in Statement :
formal-sum-add: x + y,
formal-sum: formal-sum(K;S),
formal-sum-mul: k * x,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
implies: P ⇒ Q,
infix_ap: x f y,
all: ∀x:A. B[x],
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
rng: Rng,
crng: CRng,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
formal-sum: formal-sum(K;S),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
basic-formal-sum: basic-formal-sum(K;S),
subtype_rel: A ⊆r B,
it: ⋅,
nil: [],
empty-bag: {},
map: map(f;as),
bag-map: bag-map(f;bs),
formal-sum-mul: k * x,
list_ind: list_ind,
append: as @ bs,
bag-append: as + bs,
formal-sum-add: x + y,
top: Top,
cand: A c∧ B,
exists: ∃x:A. B[x],
or: P ∨ Q,
bfs-reduce: bfs-reduce(K;S;as;bs)
Lemmas referenced :
crng_wf,
rng_car_wf,
formal-sum_wf,
equal-wf-base,
equal_wf,
implies-bfs-equiv,
formal-sum-add_wf1,
rng_plus_wf,
formal-sum-mul_wf1,
bfs-equiv-rel,
bfs-equiv_wf,
basic-formal-sum_wf,
quotient-member-eq,
formal-sum-mul_functionality,
zero-bfs_wf,
exists_wf,
bag_wf,
empty-bag_wf,
bag-append_wf,
infix_ap_wf,
empty_bag_append_lemma
Rules used in proof :
universeEquality,
axiomEquality,
isect_memberEquality,
productEquality,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
independent_functionElimination,
applyEquality,
dependent_functionElimination,
independent_isectElimination,
lambdaEquality,
hypothesisEquality,
cumulativity,
hypothesis,
rename,
setElimination,
isectElimination,
extract_by_obid,
thin,
productElimination,
pertypeElimination,
sqequalRule,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
voidEquality,
voidElimination,
dependent_pairFormation,
inrFormation
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[k,b:|K|]. \mforall{}[x:formal-sum(K;S)]. (k +K b * x = k * x + b * x)
Date html generated:
2018_05_22-PM-09_46_03
Last ObjectModification:
2018_01_09-PM-06_08_49
Theory : linear!algebra
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