Nuprl Lemma : rng_lsum_functionality_wrt_permutation
∀[r:Rng]. ∀[A:Type]. ∀[f:A ⟶ |r|]. ∀[as,bs:A List].
Σ{r} x ∈ as. f[x] = Σ{r} x ∈ bs. f[x] ∈ |r| supposing permutation(A;as;bs)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
permutation: permutation(T;L1;L2),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
so_apply: x[s],
comm: Comm(T;op),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
infix_ap: x f y,
assoc: Assoc(T;op),
so_apply: x[s1;s2],
rng: Rng,
rng_lsum: Σ{r} x ∈ as. f[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
permutation_wf,
map_wf,
rng_plus_comm,
iff_weakening_equal,
infix_ap_wf,
rng_plus_assoc,
true_wf,
squash_wf,
equal_wf,
rng_zero_wf,
rng_plus_wf,
rng_car_wf,
reduce-permutation,
permutation-map
Rules used in proof :
functionExtensionality,
cumulativity,
axiomEquality,
isect_memberEquality,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
applyEquality,
sqequalRule,
independent_isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination
Latex:
\mforall{}[r:Rng]. \mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} |r|]. \mforall{}[as,bs:A List].
\mSigma{}\{r\} x \mmember{} as. f[x] = \mSigma{}\{r\} x \mmember{} bs. f[x] supposing permutation(A;as;bs)
Date html generated:
2018_05_21-PM-09_32_43
Last ObjectModification:
2017_12_11-PM-00_45_19
Theory : matrices
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