Nuprl Lemma : p-mul-1
∀[p:{2...}]. ∀[a:p-adics(p)]. (1(p) * a = a ∈ p-adics(p))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
int_upper: {i...},
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
rng: Rng,
p-adic-ring: ℤ(p),
ring_p: IsRing(T;plus;zero;neg;times;one),
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
rng_times: *,
rng_one: 1,
monoid_p: IsMonoid(T;op;id),
group_p: IsGroup(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
ident: Ident(T;op;id),
assoc: Assoc(T;op),
inverse: Inverse(T;op;id;inv),
infix_ap: x f y,
comm: Comm(T;op),
and: P ∧ Q,
all: ∀x:A. B[x],
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
p-adics: p-adics(p),
so_lambda: λ2x.t[x],
subtract: n - m,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
so_apply: x[s]
Lemmas referenced :
p-adic-ring_wf,
crng_properties,
rng_properties,
p-adic-property,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
p-mul_wf,
p-int_wf,
nat_plus_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
nat_plus_wf,
p-adics_wf,
all_wf,
eqmod_wf,
exp_wf2,
nat_plus_subtype_nat,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_seg_wf,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_upper_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
productElimination,
lambdaFormation,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
minusEquality
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[a:p-adics(p)]. (1(p) * a = a)
Date html generated:
2018_05_21-PM-03_21_06
Last ObjectModification:
2018_05_19-AM-08_18_22
Theory : rings_1
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