Nuprl Lemma : formal-sum-mul-linear
∀[S:Type]. ∀[K:CRng]. ∀[k:|K|]. ∀[x,y:formal-sum(K;S)]. (k * x + y = k * x + k * y ∈ 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],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
and: P ∧ Q,
quotient: x,y:A//B[x; y],
formal-sum: formal-sum(K;S),
top: Top,
uimplies: b supposing a,
rng: Rng,
crng: CRng,
subtype_rel: A ⊆r B,
member: t ∈ T,
basic-formal-sum: basic-formal-sum(K;S),
formal-sum-add: x + y,
formal-sum-mul: k * x,
uall: ∀[x:A]. B[x]
Lemmas referenced :
crng_wf,
formal-sum_wf,
equal-wf-base,
formal-sum-add_functionality,
iff_weakening_equal,
rng_sig_wf,
true_wf,
squash_wf,
formal-sum-add_wf1,
formal-sum-mul_wf1,
bfs-equiv-rel,
bfs-equiv_wf,
basic-formal-sum_wf,
quotient-member-eq,
bag_wf,
rng_times_wf,
infix_ap_wf,
bag-map_wf,
bag-append_wf,
rng_car_wf,
top_wf,
subtype_rel_bag,
bag-map-append,
formal-sum-mul_functionality
Rules used in proof :
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
pertypeElimination,
pointwiseFunctionalityForEquality,
axiomEquality,
independent_pairEquality,
productElimination,
because_Cache,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaEquality,
independent_isectElimination,
cumulativity,
rename,
setElimination,
productEquality,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[k:|K|]. \mforall{}[x,y:formal-sum(K;S)]. (k * x + y = k * x + k * y)
Date html generated:
2018_05_22-PM-09_45_49
Last ObjectModification:
2018_01_09-PM-00_16_47
Theory : linear!algebra
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