Nuprl Lemma : rng_sum_single
∀[r:Rng]. ∀[i,j:ℤ]. ∀[E:{i..j-} ⟶ |r|]. ((Σ(r) i ≤ k < j. E[k]) = E[i] ∈ |r|) supposing j = (i + 1) ∈ ℤ
Proof
Definitions occuring in Statement :
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
true: True,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
subtype_rel: A ⊆r B,
rng: Rng,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
infix_ap: x f y,
so_lambda: λ2x.t[x],
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rng_plus_zero,
lelt_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
rng_plus_wf,
rng_wf,
int_subtype_base,
equal-wf-base,
rng_car_wf,
int_seg_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
rng_sum_unroll_lo,
equal_wf,
squash_wf,
true_wf,
rng_sum_unroll_base,
iff_weakening_equal
Rules used in proof :
productElimination,
equalitySymmetry,
dependent_set_memberEquality,
functionExtensionality,
because_Cache,
baseClosed,
closedConclusion,
baseApply,
applyEquality,
axiomEquality,
rename,
setElimination,
functionEquality,
equalityTransitivity,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
unionElimination,
dependent_functionElimination,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[r:Rng]. \mforall{}[i,j:\mBbbZ{}]. \mforall{}[E:\{i..j\msupminus{}\} {}\mrightarrow{} |r|]. ((\mSigma{}(r) i \mleq{} k < j. E[k]) = E[i]) supposing j = (i + 1)
Date html generated:
2018_05_21-PM-03_15_01
Last ObjectModification:
2017_12_12-AM-11_39_54
Theory : rings_1
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