Nuprl Lemma : qsum_product
∀[a,b,c,d:ℤ]. ∀[x:{a..b + 1-} ⟶ ℚ]. ∀[y:{c..d + 1-} ⟶ ℚ].
((Σa ≤ i < b. x[i] * Σc ≤ j < d. y[j]) = Σa ≤ i < b. Σc ≤ j < d. x[i] * y[j] ∈ ℚ)
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qmul: r * s,
rationals: ℚ,
int_seg: {i..j-},
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
rationals_wf,
qmul_com,
qsum_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
istype-le,
istype-less_than,
int_seg_wf,
qmul_wf,
subtype_rel_self,
iff_weakening_equal,
qsum-linearity1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
sqequalRule,
dependent_set_memberEquality_alt,
setElimination,
rename,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
addEquality,
productIsType,
because_Cache,
imageMemberEquality,
baseClosed,
functionIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[a,b,c,d:\mBbbZ{}]. \mforall{}[x:\{a..b + 1\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[y:\{c..d + 1\msupminus{}\} {}\mrightarrow{} \mBbbQ{}].
((\mSigma{}a \mleq{} i < b. x[i] * \mSigma{}c \mleq{} j < d. y[j]) = \mSigma{}a \mleq{} i < b. \mSigma{}c \mleq{} j < d. x[i] * y[j])
Date html generated:
2020_05_20-AM-09_26_05
Last ObjectModification:
2019_12_31-PM-04_59_40
Theory : rationals
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