Nuprl Lemma : mod_times_mssum_l
∀s:DSet. ∀r:Rng. ∀m:algebra{i:l}(r). ∀f:|s| ⟶ m.car. ∀u:m.car. ∀a:MSet{s}.
((u m.times (Σm x ∈ a. f[x])) = (Σm x ∈ a. (u m.times f[x])) ∈ m.car)
Proof
Definitions occuring in Statement :
mod_mssum: mod_mssum,
mset: MSet{s},
algebra: algebra{i:l}(A),
alg_times: a.times,
alg_car: a.car,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
rng: Rng,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
mod_mssum: mod_mssum,
grp_of_module: m↓grp,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
rng_of_alg: a↓rg,
rng_car: |r|,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rng: Rng,
algebra: algebra{i:l}(A),
module: A-Module,
dset: DSet,
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
infix_ap: x f y,
monoid_hom: MonHom(M1,M2),
guard: {T},
grp_op: *,
pi2: snd(t),
rng_plus: +r,
squash: ↓T,
and: P ∧ Q,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
mset_wf,
alg_car_wf,
rng_car_wf,
set_car_wf,
algebra_wf,
rng_wf,
dset_wf,
dist_hom_over_mset_for,
grp_of_module_wf2,
subtype_rel_sets,
grp_sig_wf,
monoid_p_wf,
grp_car_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
alg_times_wf,
monoid_hom_p_wf,
grp_of_module_wf,
grp_hom_formation,
grp_subtype_igrp,
abgrp_subtype_grp,
subtype_rel_transitivity,
abgrp_wf,
grp_wf,
igrp_wf,
equal_wf,
squash_wf,
true_wf,
algebra_bilinear,
infix_ap_wf,
alg_plus_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
setElimination,
rename,
functionEquality,
applyEquality,
instantiate,
setEquality,
cumulativity,
because_Cache,
lambdaEquality,
independent_isectElimination,
functionExtensionality,
dependent_set_memberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}m:algebra\{i:l\}(r). \mforall{}f:|s| {}\mrightarrow{} m.car. \mforall{}u:m.car. \mforall{}a:MSet\{s\}.
((u m.times (\mSigma{}m x \mmember{} a. f[x])) = (\mSigma{}m x \mmember{} a. (u m.times f[x])))
Date html generated:
2017_10_01-AM-10_01_01
Last ObjectModification:
2017_03_03-PM-01_03_37
Theory : list_3
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