Nuprl Lemma : p-unit-part-unique
∀[p:{2...}]. ∀[k,j:ℕ]. ∀[a,b:p-units(p)].
(a = b ∈ p-units(p)) ∧ (k = j ∈ ℤ) supposing p^k(p) * a = p^j(p) * b ∈ p-adics(p)
Proof
Definitions occuring in Statement :
p-units: p-units(p),
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
exp: i^n,
int_upper: {i...},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
int_upper: {i...},
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
all: ∀x:A. B[x],
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
p-units: p-units(p),
sq_type: SQType(T),
subtract: n - m,
label: ...$L... t,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
guard: {T},
prop: ℙ,
int_seg: {i..j-},
less_than: a < b,
p-adics: p-adics(p),
p-mul: x * y,
p-int: k(p),
p-reduce: i mod(p^n),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o
Lemmas referenced :
sq_stable__and,
equal_wf,
p-units_wf,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
sq_stable__equal,
p-adics_wf,
p-mul_wf,
subtype_rel_sets_simple,
less_than_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
istype-le,
p-int_wf,
exp_wf2,
istype-nat,
istype-int_upper,
nat_wf,
iff_weakening_equal,
subtype_rel_self,
add-zero,
zero-mul,
add-mul-special,
add-associates,
add-swap,
minus-minus,
minus-add,
minus-one-mul-top,
minus-one-mul,
condition-implies-le,
less-iff-le,
exp-positive,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermMultiply_wf,
intformeq_wf,
multiply-is-int-iff,
decidable__equal_int,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
int_upper_properties,
nat_properties,
subtract_wf,
exp_add,
false_wf,
exp_wf_nat_plus,
true_wf,
squash_wf,
nat_plus_wf,
subtype_base_sq,
p-mul-int,
p-mul-assoc,
p-mul-int-cancelation-1,
int_seg_wf,
exp1,
rem-exact,
exp_step,
rem_bounds_1,
nat_plus_properties,
nequal_wf,
subtype_rel_sets,
upper_subtype_nat,
nat_plus_subtype_nat,
exp_wf4,
modulus-is-rem,
p-reduce-0,
istype-universe,
istype-less_than,
itermAdd_wf,
int_term_value_add_lemma,
p-adic-property,
eqmod_wf,
equal-wf-T-base,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
isect_memberEquality_alt,
intEquality,
applyEquality,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
equalityIstype,
inhabitedIsType,
independent_functionElimination,
lambdaFormation_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
independent_pairFormation,
productElimination,
unionElimination,
voidElimination,
Error :memTop,
minusEquality,
addEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
lambdaFormation,
dependent_set_memberEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
lambdaEquality,
cumulativity,
instantiate,
setEquality,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
functionEquality,
multiplyEquality,
functionIsType
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[k,j:\mBbbN{}]. \mforall{}[a,b:p-units(p)]. (a = b) \mwedge{} (k = j) supposing p\^{}k(p) * a = p\^{}j(p) * b
Date html generated:
2020_05_19-PM-10_08_30
Last ObjectModification:
2020_01_08-PM-06_00_00
Theory : rings_1
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