Nuprl Lemma : formal-sum-mul-mul
∀[S:Type]. ∀[K:CRng]. ∀[k,b:|K|]. ∀[x:formal-sum(K;S)]. (k * b * x = k * b * x ∈ formal-sum(K;S))
Proof
Definitions occuring in Statement :
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_times: *,
rng_car: |r|
Definitions unfolded in proof :
rng: Rng,
crng: CRng,
compose: f o g,
top: Top,
member: t ∈ T,
basic-formal-sum: basic-formal-sum(K;S),
formal-sum-mul: k * x,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
infix_ap: x f y,
all: ∀x:A. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
quotient: x,y:A//B[x; y],
formal-sum: formal-sum(K;S)
Lemmas referenced :
crng_wf,
rng_car_wf,
bag_wf,
bag-map-map,
iff_weakening_equal,
rng_times_wf,
infix_ap_wf,
rng_times_assoc,
equal_wf,
true_wf,
squash_wf,
bag-map_wf,
formal-sum_wf,
equal-wf-base,
formal-sum-mul_wf1,
bfs-equiv-rel,
bfs-equiv_wf,
basic-formal-sum_wf,
quotient-member-eq,
rng_sig_wf,
formal-sum-mul_functionality
Rules used in proof :
universeEquality,
because_Cache,
cumulativity,
hypothesisEquality,
rename,
setElimination,
productEquality,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_pairEquality,
productElimination,
functionExtensionality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
dependent_functionElimination,
pertypeElimination,
pointwiseFunctionalityForEquality
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[k,b:|K|]. \mforall{}[x:formal-sum(K;S)]. (k * b * x = k * b * x)
Date html generated:
2018_05_22-PM-09_45_52
Last ObjectModification:
2018_01_09-PM-00_59_13
Theory : linear!algebra
Home
Index