Nuprl Lemma : rng_times_sum_l
∀[r:Rng]. ∀[a,b:ℤ].
∀[E:{a..b-} ⟶ |r|]. ∀[u:|r|]. ((u * (Σ(r) a ≤ j < b. E[j])) = (Σ(r) a ≤ j < b. u * E[j]) ∈ |r|) supposing a ≤ b
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_times: *,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rng: Rng,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
guard: {T},
int_seg: {i..j-},
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
lelt: i ≤ j < k
Lemmas referenced :
rng_car_wf,
int_seg_wf,
le_wf,
rng_wf,
int_le_to_int_upper_uniform,
uall_wf,
equal_wf,
infix_ap_wf,
rng_times_wf,
rng_sum_wf,
int_upper_wf,
int_upper_ind_uniform,
decidable__equal_int,
squash_wf,
true_wf,
rng_sum_unroll_base,
iff_weakening_equal,
rng_times_zero,
rng_zero_wf,
rng_sum_unroll_hi,
int_upper_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rng_plus_wf,
subtract_wf,
itermSubtract_wf,
itermConstant_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
lelt_wf,
decidable__le,
rng_times_over_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
setElimination,
rename,
functionEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
productElimination,
independent_functionElimination,
instantiate,
lambdaFormation,
unionElimination,
imageElimination,
universeEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} |r|]. \mforall{}[u:|r|]. ((u * (\mSigma{}(r) a \mleq{} j < b. E[j])) = (\mSigma{}(r) a \mleq{} j < b. u * E[j]))
supposing a \mleq{} b
Date html generated:
2017_10_01-AM-08_19_27
Last ObjectModification:
2017_02_28-PM-02_04_15
Theory : rings_1
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