Nuprl Lemma : rng_mssum_of_plus
∀r:Rng. ∀s:DSet. ∀e,f:|s| ⟶ |r|. ∀a:MSet{s}. ((Σx ∈ a. (e[x] +r f[x])) = ((Σx ∈ a. e[x]) +r (Σx ∈ a. f[x])) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_mssum: rng_mssum,
mset: MSet{s},
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,
rng_plus: +r,
rng_car: |r|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
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,
rng_mssum: rng_mssum,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t)
Lemmas referenced :
mset_for_of_op,
add_grp_of_rng_wf_b,
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,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
setEquality,
cumulativity,
setElimination,
rename,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}r:Rng. \mforall{}s:DSet. \mforall{}e,f:|s| {}\mrightarrow{} |r|. \mforall{}a:MSet\{s\}.
((\mSigma{}x \mmember{} a. (e[x] +r f[x])) = ((\mSigma{}x \mmember{} a. e[x]) +r (\mSigma{}x \mmember{} a. f[x])))
Date html generated:
2016_05_16-AM-08_12_01
Last ObjectModification:
2015_12_28-PM-06_06_26
Theory : list_3
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