Nuprl Lemma : mset_for_functionality
∀s:DSet. ∀g:IAbMonoid. ∀f,f':|s| ⟶ |g|. ∀a,a':MSet{s}.
((a = a' ∈ MSet{s})
⇒ (∀x:|s|. ((↑(x ∈b a)) ⇒ (f[x] = f'[x] ∈ |g|)))
⇒ ((msFor{g} x ∈ a. f[x]) = (msFor{g} x ∈ a'. f'[x]) ∈ |g|))
Proof
Definitions occuring in Statement :
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_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],
sq_stable: SqStable(P),
mset: MSet{s},
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
mset_for: mset_for,
mset_mem: mset_mem,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True
Lemmas referenced :
all_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
equal_wf,
grp_car_wf,
mset_wf,
iabmonoid_wf,
dset_wf,
sq_stable__all,
mset_for_wf,
sq_stable__equal,
squash_wf,
list_wf,
permr_wf,
equal-wf-base,
mem_wf,
mon_for_functionality_wrt_permr,
permr_weakening,
mem_f_wf,
mem_iff_mem_f,
true_wf
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,
equalityTransitivity,
independent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productEquality,
imageElimination,
natural_numberEquality
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f,f':|s| {}\mrightarrow{} |g|. \mforall{}a,a':MSet\{s\}.
((a = a')
{}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} a)) {}\mRightarrow{} (f[x] = f'[x])))
{}\mRightarrow{} ((msFor\{g\} x \mmember{} a. f[x]) = (msFor\{g\} x \mmember{} a'. f'[x])))
Date html generated:
2017_10_01-AM-09_59_17
Last ObjectModification:
2017_03_03-PM-01_00_09
Theory : mset
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