Nuprl Lemma : p-inv_wf
∀p:{p:{2...}| prime(p)} . ∀a:p-units(p). (p-inv(p;a) ∈ {b:p-adics(p)| a * b = 1(p) ∈ p-adics(p)} )
Proof
Definitions occuring in Statement :
p-inv: p-inv(p;a),
p-units: p-units(p),
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
prime: prime(a),
int_upper: {i...},
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
p-units: p-units(p),
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_upper: {i...},
not: ¬A,
implies: P ⇒ Q,
p-adics: p-adics(p),
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
le: A ≤ B,
false: False,
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
p-inv: p-inv(p;a),
has-value: (a)↓,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
p-adic-inv-lemma,
sign: sign(x),
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_exists: ∃x:A [B[x]],
sq_stable: SqStable(P),
eqmod: a ≡ b mod m,
divides: b | a,
ge: i ≥ j ,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
p-int: k(p),
p-mul: x * y
Lemmas referenced :
p-adics_wf,
istype-int,
istype-less_than,
set_subtype_base,
lelt_wf,
exp_wf2,
istype-false,
istype-le,
int_subtype_base,
istype-void,
istype-int_upper,
prime_wf,
value-type-has-value,
nat_plus_wf,
set-value-type,
less_than_wf,
int-value-type,
exp_wf_nat_plus,
nat_plus_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
decidable__lt,
int_seg_wf,
p-adic-inv-lemma,
subtype_base_sq,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
int_seg_properties,
nat_plus_subtype_nat,
intformeq_wf,
int_formula_prop_eq_lemma,
exp-positive,
absval_pos,
subtype_rel_self,
int_upper_wf,
not_wf,
equal-wf-base,
sq_exists_wf,
eqmod_wf,
itermAdd_wf,
int_term_value_add_lemma,
sq_stable__eqmod,
p-adic-property,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mul_assoc,
iff_weakening_equal,
exp_add,
exp1,
int_seg_subtype_nat,
eqmod_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening,
nat_properties,
decidable__equal_int,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
subtract_wf,
divides_wf,
divides_subtract,
divides_product,
exp_step,
divisor_bound,
subtype_rel_set,
nat_wf,
upper_subtype_nat,
prime-power-divides-product,
sq_stable_from_decidable,
decidable__prime,
mul_cancel_in_eq,
exp_wf3,
int_nzero_wf,
subtype_rel_sets_simple,
nequal_wf,
itermMinus_wf,
int_term_value_minus_lemma,
p-mul_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
p-int_wf,
p-adics-equal,
p-reduce_wf,
mul-commutes,
p-reduce-eqmod
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
setIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
functionIsType,
equalityIstype,
applyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
intEquality,
lambdaEquality_alt,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
functionExtensionality,
callbyvalueReduce,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
because_Cache,
instantiate,
cumulativity,
inhabitedIsType,
productElimination,
equalityElimination,
equalityTransitivity,
promote_hyp,
applyLambdaEquality,
imageElimination,
functionEquality,
setEquality,
multiplyEquality,
addEquality,
universeEquality,
baseApply,
closedConclusion,
inrFormation_alt,
inlFormation_alt,
minusEquality
Latex:
\mforall{}p:\{p:\{2...\}| prime(p)\} . \mforall{}a:p-units(p). (p-inv(p;a) \mmember{} \{b:p-adics(p)| a * b = 1(p)\} )
Date html generated:
2019_10_15-AM-10_35_07
Last ObjectModification:
2019_06_24-PM-00_57_15
Theory : rings_1
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