Nuprl Lemma : rng_times_nat_op
∀[r:Rng]. ∀[a,b:|r|]. ∀[n:ℕ]. ((a * (n ⋅r b)) = (n ⋅r (a * b)) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_nat_op: n ⋅r e,
rng: Rng,
rng_times: *,
rng_car: |r|,
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rng_nat_op: n ⋅r e,
mon_nat_op: n ⋅ e,
nat_op: n x(op;id) e,
mon_itop: Π lb ≤ i < ub. E[i],
rng_sum: rng_sum,
rng: Rng,
nat: ℕ,
uimplies: b supposing a,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
int_seg_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
rng_times_sum_l,
rng_wf,
rng_car_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[r:Rng]. \mforall{}[a,b:|r|]. \mforall{}[n:\mBbbN{}]. ((a * (n \mcdot{}r b)) = (n \mcdot{}r (a * b)))
Date html generated:
2016_05_15-PM-00_28_27
Last ObjectModification:
2016_01_15-AM-08_51_07
Theory : rings_1
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