Nuprl Lemma : formal-sum-mul-0
∀[S:Type]. ∀[K:Rng]. ∀[x:formal-sum(K;S)]. (0 * 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],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_zero: 0,
empty-bag: {}
Definitions unfolded in proof :
prop: ℙ,
basic-formal-sum: basic-formal-sum(K;S),
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
all: ∀x:A. B[x],
rng: Rng,
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],
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
top: Top,
or: P ∨ Q,
bfs-reduce: bfs-reduce(K;S;as;bs),
exists: ∃x:A. B[x],
true: True,
pi2: snd(t),
compose: f o g,
squash: ↓T,
formal-sum-mul: k * x,
zero-bfs: 0 * ss
Lemmas referenced :
rng_wf,
formal-sum_wf,
equal-wf-base,
equal_wf,
implies-bfs-equiv,
rng_car_wf,
empty-bag_wf,
rng_zero_wf,
formal-sum-mul_wf1,
bfs-equiv-rel,
bfs-equiv_wf,
quotient-member-eq,
basic-formal-sum_wf,
equal-wf-base-T,
bag_wf,
rng_plus_wf,
infix_ap_wf,
bag-append_wf,
exists_wf,
empty_bag_append_lemma,
zero-bfs_wf,
pi2_wf,
bag-map_wf,
rng_times_zero,
true_wf,
squash_wf,
bag-map-map
Rules used in proof :
universeEquality,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productEquality,
dependent_functionElimination,
independent_isectElimination,
lambdaEquality,
lambdaFormation,
hypothesisEquality,
cumulativity,
rename,
setElimination,
isectElimination,
extract_by_obid,
equalitySymmetry,
hypothesis,
equalityTransitivity,
thin,
productElimination,
pertypeElimination,
sqequalRule,
because_Cache,
pointwiseFunctionalityForEquality,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
applyEquality,
voidEquality,
voidElimination,
inlFormation,
independent_pairEquality,
dependent_pairFormation,
imageMemberEquality,
natural_numberEquality,
functionExtensionality,
functionEquality,
imageElimination
Latex:
\mforall{}[S:Type]. \mforall{}[K:Rng]. \mforall{}[x:formal-sum(K;S)]. (0 * x = \{\})
Date html generated:
2018_05_22-PM-09_45_46
Last ObjectModification:
2018_01_09-PM-01_12_42
Theory : linear!algebra
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