Nuprl Lemma : rng_mssum_functionality_wrt_bsubmset
∀s:DSet. ∀r:Rng. ∀f,f':|s| ⟶ |r|. ∀p,q:MSet{s}.
((∀x:|s|. ((↑(x ∈b q - p)) ⇒ (f'[x] = 0 ∈ |r|)))
⇒ (↑(p ⊆b q))
⇒ (∀x:|s|. ((↑(x ∈b p)) ⇒ (f[x] = f'[x] ∈ |r|)))
⇒ ((Σx ∈ p. f[x]) = (Σx ∈ q. f'[x]) ∈ |r|))
Proof
Definitions occuring in Statement :
rng_mssum: rng_mssum,
bsubmset: a ⊆b b,
mset_diff: a - b,
mset_mem: mset_mem,
mset: MSet{s},
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,
rng_zero: 0,
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_id: e,
pi2: snd(t)
Lemmas referenced :
mset_for_functionality_wrt_bsubmset,
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,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
setEquality,
cumulativity,
setElimination,
rename,
lambdaEquality,
independent_isectElimination
Latex:
\mforall{}s:DSet. \mforall{}r:Rng. \mforall{}f,f':|s| {}\mrightarrow{} |r|. \mforall{}p,q:MSet\{s\}.
((\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} q - p)) {}\mRightarrow{} (f'[x] = 0)))
{}\mRightarrow{} (\muparrow{}(p \msubseteq{}\msubb{} q))
{}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} p)) {}\mRightarrow{} (f[x] = f'[x])))
{}\mRightarrow{} ((\mSigma{}x \mmember{} p. f[x]) = (\mSigma{}x \mmember{} q. f'[x])))
Date html generated:
2016_05_16-AM-08_11_52
Last ObjectModification:
2015_12_28-PM-06_06_24
Theory : list_3
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