Nuprl Lemma : rng_times_lsum_l
∀r:Rng. ∀A:Type. ∀as:A List. ∀f:A ⟶ |r|. ∀u:|r|. ((u * (Σ{A,r} x ∈ as. f[x])) = (Σ{A,r} x ∈ as. (u * f[x])) ∈ |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|.
((u * (\mSigma{}\{A,r\} x \mmember{} as. f[x])) = (\mSigma{}\{A,r\} x \mmember{} as. (u * f[x])))
Date html generated:
2017_10_01-AM-10_00_57
Last ObjectModification:
2017_03_03-PM-01_02_14
Theory : list_3
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