Nuprl Lemma : rng_times_lsum_r
∀r:Rng. ∀A:Type. ∀as:A List. ∀f:A ⟶ |r|. ∀u:|r|.  (((Σ{A,r} x ∈ as. f[x]) * u) = (Σ{A,r} x ∈ as. (f[x] * u)) ∈ |r|)
Proof
Definitions occuring in Statement : 
rng_lsum: Σ{A,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 : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
rng: Rng
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
rng_lsum: Σ{A,r} x ∈ as. f[x]
, 
top: Top
, 
add_grp_of_rng: r↓+gp
, 
grp_id: e
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
and: P ∧ Q
, 
grp_op: *
, 
squash: ↓T
, 
prop: ℙ
, 
infix_ap: x f y
, 
true: True
, 
subtype_rel: A ⊆r B
, 
uimplies: b supposing a
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
Lemmas referenced : 
list_induction, 
all_wf, 
rng_car_wf, 
equal_wf, 
infix_ap_wf, 
rng_times_wf, 
rng_lsum_wf, 
list_wf, 
mon_for_nil_lemma, 
rng_times_zero, 
mon_for_cons_lemma, 
squash_wf, 
true_wf, 
rng_plus_wf, 
iff_weakening_equal, 
rng_times_over_plus, 
rng_plus_comm, 
rng_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
thin, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
functionEquality, 
cumulativity, 
setElimination, 
rename, 
because_Cache, 
hypothesis, 
dependent_functionElimination, 
applyEquality, 
functionExtensionality, 
independent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
productElimination, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
independent_isectElimination
Latex:
\mforall{}r:Rng.  \mforall{}A:Type.  \mforall{}as:A  List.  \mforall{}f:A  {}\mrightarrow{}  |r|.  \mforall{}u:|r|.
    (((\mSigma{}\{A,r\}  x  \mmember{}  as.  f[x])  *  u)  =  (\mSigma{}\{A,r\}  x  \mmember{}  as.  (f[x]  *  u)))
Date html generated:
2017_10_01-AM-10_00_58
Last ObjectModification:
2017_03_03-PM-01_02_13
Theory : list_3
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