Nuprl Lemma : fset_for_when_eq
∀s:DSet. ∀g:IAbMonoid. ∀f:|s| ⟶ |g|. ∀e:|s|. ∀as:FiniteSet{s}.
((↑(e ∈b as)) ⇒ ((msFor{g} x ∈ as. when x (=b) e. f[x]) = f[e] ∈ |g|))
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset_mem: mset_mem,
finite_set: FiniteSet{s},
assert: ↑b,
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
mon_when: when b. p,
iabmonoid: IAbMonoid,
grp_car: |g|,
dset: DSet,
set_eq: =b,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
so_lambda: λ2x.t[x],
so_apply: x[s],
dset: DSet,
infix_ap: x f y,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
finite_set: FiniteSet{s},
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
grp_car_wf,
fset_for_when_unique,
set_car_wf,
set_eq_wf,
assert_of_dset_eq,
assert_wf,
mset_mem_wf,
iff_weakening_equal,
finite_set_wf,
iabmonoid_wf,
dset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
setElimination,
rename,
dependent_functionElimination,
sqequalRule,
functionExtensionality,
because_Cache,
independent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
functionEquality
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}e:|s|. \mforall{}as:FiniteSet\{s\}.
((\muparrow{}(e \mmember{}\msubb{} as)) {}\mRightarrow{} ((msFor\{g\} x \mmember{} as. when x (=\msubb{}) e. f[x]) = f[e]))
Date html generated:
2017_10_01-AM-10_00_43
Last ObjectModification:
2017_03_03-PM-01_01_59
Theory : mset
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