Nuprl Lemma : null-formal-sum-mul
∀[S:Type]. ∀K:CRng. ∀[x:basic-formal-sum(K;S)]. ∀k:|K|. (null-formal-sum(K;S;x) ⇒ null-formal-sum(K;S;k * x))
Proof
Definitions occuring in Statement :
null-formal-sum: null-formal-sum(K;S;fs),
formal-sum-mul: k * x,
basic-formal-sum: basic-formal-sum(K;S),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
subtype_rel: A ⊆r B,
basic-formal-sum: basic-formal-sum(K;S),
rng: Rng,
crng: CRng,
member: t ∈ T,
exists: ∃x:A. B[x],
null-formal-sum: null-formal-sum(K;S;fs),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
infix_ap: x f y,
squash: ↓T,
compose: f o g,
top: Top,
uimplies: b supposing a,
neg-bfs: -(fs),
formal-sum-mul: k * x,
zero-bfs: 0 * ss,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
crng_wf,
basic-formal-sum_wf,
null-formal-sum_wf,
exists_wf,
zero-bfs_wf,
neg-bfs_wf,
bag-append_wf,
rng_car_wf,
bag_wf,
equal_wf,
formal-sum-mul_wf1,
rng_times_zero,
rng_times_wf,
true_wf,
squash_wf,
bag-map_wf,
subtype_rel_bag,
top_wf,
rng_sig_wf,
iff_weakening_equal,
bag-map-append,
bag-map-map,
rng_times_over_minus
Rules used in proof :
universeEquality,
productEquality,
lambdaEquality,
applyEquality,
sqequalRule,
hypothesis,
because_Cache,
rename,
setElimination,
hypothesisEquality,
cumulativity,
isectElimination,
extract_by_obid,
introduction,
cut,
dependent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionEquality,
baseClosed,
imageMemberEquality,
independent_pairEquality,
spreadEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
functionExtensionality,
voidEquality,
voidElimination,
isect_memberEquality,
independent_isectElimination,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
independent_functionElimination
Latex:
\mforall{}[S:Type]
\mforall{}K:CRng
\mforall{}[x:basic-formal-sum(K;S)]. \mforall{}k:|K|. (null-formal-sum(K;S;x) {}\mRightarrow{} null-formal-sum(K;S;k * x))
Date html generated:
2018_05_22-PM-09_47_31
Last ObjectModification:
2018_01_09-PM-00_47_11
Theory : linear!algebra
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