Nuprl Lemma : rng_lsum_map
∀[r:Rng]. ∀[A,B:Type]. ∀[g:A ⟶ B]. ∀f:B ⟶ |r|. ∀as:A List. (Σ{r} x ∈ map(g;as). f[x] = Σ{r} x ∈ as. f[g x] ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
infix_ap: x f y,
prop: ℙ,
and: P ∧ Q,
top: Top,
implies: P ⇒ Q,
so_apply: x[s],
rng: Rng,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
rng_plus_wf,
and_wf,
rng_lsum_cons_lemma,
map_cons_lemma,
rng_zero_wf,
rng_lsum_nil_lemma,
map_nil_lemma,
list_wf,
map_wf,
rng_lsum_wf,
rng_car_wf,
equal_wf,
list_induction
Rules used in proof :
universeEquality,
axiomEquality,
functionEquality,
because_Cache,
productElimination,
applyLambdaEquality,
equalityTransitivity,
independent_pairFormation,
dependent_set_memberEquality,
equalitySymmetry,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
cumulativity,
hypothesis,
rename,
setElimination,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[r:Rng]. \mforall{}[A,B:Type]. \mforall{}[g:A {}\mrightarrow{} B].
\mforall{}f:B {}\mrightarrow{} |r|. \mforall{}as:A List. (\mSigma{}\{r\} x \mmember{} map(g;as). f[x] = \mSigma{}\{r\} x \mmember{} as. f[g x])
Date html generated:
2018_05_21-PM-09_32_59
Last ObjectModification:
2017_12_11-PM-01_12_37
Theory : matrices
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