Nuprl Lemma : rng_prod_const_mul
∀[r:CRng]. ∀[c:|r|]. ∀[n:ℕ]. ∀[F:ℕn ⟶ |r|]. ((Π(r) 0 ≤ i < n. F[i] * c) = ((c ↑r n) * (Π(r) 0 ≤ i < n. F[i])) ∈ |r|)
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
rng_nexp: e ↑r n,
rng_prod: rng_prod,
crng: CRng,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
crng: CRng,
rng: Rng,
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
infix_ap: x f y,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
ringeq_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
rng_car_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
crng_wf,
rng_prod_empty_lemma,
equal_wf,
squash_wf,
true_wf,
rng_one_wf,
rng_times_one,
rng_nexp_wf,
false_wf,
le_wf,
subtype_rel_self,
iff_weakening_equal,
rng_nexp_zero,
rng_times_wf,
decidable__lt,
lelt_wf,
infix_ap_wf,
rng_prod_wf,
rng_prod_unroll_hi,
rng_nexp_unroll,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
int-to-ring_wf,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_minus_lemma,
ring_term_value_const_lemma,
int-to-ring-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionEquality,
because_Cache,
unionElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
productElimination,
imageMemberEquality,
baseClosed,
instantiate,
functionExtensionality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[c:|r|]. \mforall{}[n:\mBbbN{}]. \mforall{}[F:\mBbbN{}n {}\mrightarrow{} |r|].
((\mPi{}(r) 0 \mleq{} i < n. F[i] * c) = ((c \muparrow{}r n) * (\mPi{}(r) 0 \mleq{} i < n. F[i])))
Date html generated:
2018_05_21-PM-09_33_58
Last ObjectModification:
2018_05_19-PM-04_23_14
Theory : matrices
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