Nuprl Lemma : rng_times_lsum_r
∀r:Rng. ∀A:Type. ∀as:A List. ∀f:A ⟶ |r|. ∀u:|r|. ((Σ{r} x ∈ as. f[x] * u) = Σ{r} x ∈ as. (f[x] * u) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
list: T List,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
infix_ap: x f y,
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
top: Top,
implies: P ⇒ Q,
so_apply: x[s],
rng: Rng,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
rng_wf,
rng_plus_comm,
rng_times_over_plus,
iff_weakening_equal,
rng_plus_wf,
true_wf,
squash_wf,
rng_lsum_cons_lemma,
rng_times_zero,
rng_lsum_nil_lemma,
list_wf,
rng_lsum_wf,
rng_times_wf,
infix_ap_wf,
equal_wf,
rng_car_wf,
all_wf,
list_induction
Rules used in proof :
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
productElimination,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
hypothesis,
because_Cache,
rename,
setElimination,
cumulativity,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}r:Rng. \mforall{}A:Type. \mforall{}as:A List. \mforall{}f:A {}\mrightarrow{} |r|. \mforall{}u:|r|.
((\mSigma{}\{r\} x \mmember{} as. f[x] * u) = \mSigma{}\{r\} x \mmember{} as. (f[x] * u))
Date html generated:
2018_05_21-PM-09_32_53
Last ObjectModification:
2017_12_11-PM-04_21_51
Theory : matrices
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