Nuprl Lemma : mset_for_functionality_wrt_bsubmset
∀s:DSet. ∀g:IAbMonoid. ∀f,f':|s| ⟶ |g|. ∀p,q:MSet{s}.
((∀x:|s|. ((↑(x ∈b q - p)) ⇒ (f'[x] = e ∈ |g|)))
⇒ (↑(p ⊆b q))
⇒ (∀x:|s|. ((↑(x ∈b p)) ⇒ (f[x] = f'[x] ∈ |g|)))
⇒ ((msFor{g} x ∈ p. f[x]) = (msFor{g} x ∈ q. f'[x]) ∈ |g|))
Proof
Definitions occuring in Statement :
bsubmset: a ⊆b b,
mset_diff: a - b,
mset_for: mset_for,
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,
iabmonoid: IAbMonoid,
grp_id: e,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
dset: DSet,
so_lambda: λ2x.t[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
so_apply: x[s],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
all_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
equal_wf,
grp_car_wf,
bsubmset_wf,
mset_diff_wf,
grp_id_wf,
mset_wf,
iabmonoid_wf,
dset_wf,
mset_for_dom_shift,
squash_wf,
true_wf,
mset_for_wf,
iff_weakening_equal,
mset_for_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
applyEquality,
functionExtensionality,
because_Cache,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f,f':|s| {}\mrightarrow{} |g|. \mforall{}p,q:MSet\{s\}.
((\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} q - p)) {}\mRightarrow{} (f'[x] = e)))
{}\mRightarrow{} (\muparrow{}(p \msubseteq{}\msubb{} q))
{}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} p)) {}\mRightarrow{} (f[x] = f'[x])))
{}\mRightarrow{} ((msFor\{g\} x \mmember{} p. f[x]) = (msFor\{g\} x \mmember{} q. f'[x])))
Date html generated:
2017_10_01-AM-10_00_51
Last ObjectModification:
2017_03_03-PM-01_02_02
Theory : mset
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