Nuprl Lemma : rng_sum_plus
∀[r:Rng]. ∀[a,b:ℤ].
∀[E,F:{a..b-} ⟶ |r|]. ((Σ(r) a ≤ i < b. E[i] +r F[i]) = ((Σ(r) a ≤ i < b. E[i]) +r (Σ(r) a ≤ i < b. F[i])) ∈ |r|)
supposing a ≤ b
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_plus: +r,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rng_sum: rng_sum,
add_grp_of_rng: r↓+gp,
grp_car: |g|,
pi1: fst(t),
grp_op: *,
pi2: snd(t),
rng: Rng
Lemmas referenced :
mon_itop_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,
int_seg_wf,
rng_car_wf,
le_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
setEquality,
cumulativity,
setElimination,
rename,
because_Cache,
lambdaEquality,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
axiomEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
intEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}[E,F:\{a..b\msupminus{}\} {}\mrightarrow{} |r|].
((\mSigma{}(r) a \mleq{} i < b. E[i] +r F[i]) = ((\mSigma{}(r) a \mleq{} i < b. E[i]) +r (\mSigma{}(r) a \mleq{} i < b. F[i])))
supposing a \mleq{} b
Date html generated:
2018_05_21-PM-03_15_04
Last ObjectModification:
2018_05_19-AM-08_08_01
Theory : rings_1
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