Nuprl Lemma : mod_mssum_functionality
∀s:DSet. ∀r:Rng. ∀m:r-Module. ∀f,f':|s| ⟶ m.car. ∀a,a':MSet{s}.
((a = a' ∈ MSet{s})
⇒ (∀x:|s|. ((↑(x ∈b a)) ⇒ (f[x] = f'[x] ∈ m.car)))
⇒ ((Σm x ∈ a. f[x]) = (Σm x ∈ a'. f'[x]) ∈ m.car))
Proof
Definitions occuring in Statement :
mod_mssum: mod_mssum,
mset_mem: mset_mem,
mset: MSet{s},
module: A-Module,
alg_car: a.car,
assert: ↑b,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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_functionality,
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,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
isectElimination,
setEquality,
cumulativity,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}m:r-Module. \mforall{}f,f':|s| {}\mrightarrow{} m.car. \mforall{}a,a':MSet\{s\}.
((a = a') {}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} a)) {}\mRightarrow{} (f[x] = f'[x]))) {}\mRightarrow{} ((\mSigma{}m x \mmember{} a. f[x]) = (\mSigma{}m x \mmember{} a'. f'[x])))
Date html generated:
2016_05_16-AM-08_12_09
Last ObjectModification:
2015_12_28-PM-06_06_14
Theory : list_3
Home
Index