Nuprl Lemma : mod_mssum_swap
∀r:Rng. ∀m:r-Module. ∀s,s':DSet. ∀f:|s| ⟶ |s'| ⟶ m.car. ∀a:MSet{s}. ∀b:MSet{s'}.
((Σm x ∈ a. Σm y ∈ b. f[x;y]) = (Σm y ∈ b. Σm x ∈ a. f[x;y]) ∈ m.car)
Proof
Definitions occuring in Statement :
mod_mssum: mod_mssum,
mset: MSet{s},
module: A-Module,
alg_car: a.car,
so_apply: x[s1;s2],
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],
member: t ∈ T,
rng: Rng,
subtype_rel: A ⊆r B,
abgrp: AbGrp,
grp: Group{i},
mon: Mon,
iabmonoid: IAbMonoid,
imon: IMonoid,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
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|
Lemmas referenced :
mset_for_swap,
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,
module_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
cumulativity,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}r:Rng. \mforall{}m:r-Module. \mforall{}s,s':DSet. \mforall{}f:|s| {}\mrightarrow{} |s'| {}\mrightarrow{} m.car. \mforall{}a:MSet\{s\}. \mforall{}b:MSet\{s'\}.
((\mSigma{}m x \mmember{} a. \mSigma{}m y \mmember{} b. f[x;y]) = (\mSigma{}m y \mmember{} b. \mSigma{}m x \mmember{} a. f[x;y]))
Date html generated:
2016_05_16-AM-08_12_28
Last ObjectModification:
2015_12_28-PM-06_06_15
Theory : list_3
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