Nuprl Lemma : rng_nat_op-int
∀[n:ℕ]. ∀[a:ℤ]. (n ⋅ℤ-rng a ~ n * a)
Proof
Definitions occuring in Statement :
rng_nat_op: n ⋅r e,
int_ring: ℤ-rng,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
rng_nat_op: n ⋅r e,
mon_nat_op: n ⋅ e,
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
int_ring: ℤ-rng,
rng_plus: +r,
rng_zero: 0,
nat_op: n x(op;id) e,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
infix_ap: x f y,
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
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,
zero-mul,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
subtype_base_sq,
int_subtype_base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
unionElimination,
instantiate,
cumulativity,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
because_Cache,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbZ{}]. (n \mcdot{}\mBbbZ{}-rng a \msim{} n * a)
Date html generated:
2017_10_01-AM-08_18_50
Last ObjectModification:
2017_02_28-PM-02_03_34
Theory : rings_1
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