Nuprl Lemma : rng_minus_lsum
∀r:Rng. ∀A:Type. ∀as:A List. ∀f:A ⟶ |r|. ((-r Σ{r} x ∈ as. f[x]) = Σ{r} x ∈ as. (-r f[x]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
list: T List,
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_minus: -r,
rng_car: |r|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
rng: Rng,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
list_induction,
all_wf,
rng_car_wf,
equal_wf,
rng_minus_wf,
rng_lsum_wf,
list_wf,
rng_lsum_nil_lemma,
squash_wf,
true_wf,
rng_minus_zero,
rng_zero_wf,
iff_weakening_equal,
rng_lsum_cons_lemma,
infix_ap_wf,
rng_plus_wf,
rng_minus_over_plus,
rng_plus_comm,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
setElimination,
rename,
hypothesis,
applyEquality,
because_Cache,
functionExtensionality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination
Latex:
\mforall{}r:Rng. \mforall{}A:Type. \mforall{}as:A List. \mforall{}f:A {}\mrightarrow{} |r|. ((-r \mSigma{}\{r\} x \mmember{} as. f[x]) = \mSigma{}\{r\} x \mmember{} as. (-r f[x]))
Date html generated:
2018_05_21-PM-09_32_45
Last ObjectModification:
2017_12_11-PM-03_43_27
Theory : matrices
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