Nuprl Lemma : p-adics-equal
∀p:ℕ+. ∀x,y:p-adics(p). uiff(x = y ∈ p-adics(p);∀n:ℕ+. ((x n) ≡ (y n) mod p^n))
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
eqmod: a ≡ b mod m,
exp: i^n,
nat_plus: ℕ+,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
p-adics: p-adics(p),
int_seg: {i..j-},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
lelt: i ≤ j < k,
guard: {T},
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
equal-p-adics,
eqmod_weakening,
exp_wf2,
nat_plus_subtype_nat,
int_seg_wf,
and_wf,
equal_wf,
nat_plus_wf,
p-adics_wf,
all_wf,
eqmod_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
less_than_transitivity2,
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,
le_wf,
modulus_base,
modulus-equal-iff-eqmod,
int_seg_properties,
nat_plus_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
mod_bounds_1,
exp_wf3,
subtype_rel_sets,
nequal_wf,
equal-wf-base,
int_subtype_base,
mod_bounds,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
sqequalRule,
setElimination,
because_Cache,
lambdaEquality,
natural_numberEquality,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
functionEquality,
intEquality,
applyLambdaEquality,
functionExtensionality,
addEquality,
unionElimination,
voidElimination,
independent_functionElimination,
minusEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
setEquality,
baseClosed
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}x,y:p-adics(p). uiff(x = y;\mforall{}n:\mBbbN{}\msupplus{}. ((x n) \mequiv{} (y n) mod p\^{}n))
Date html generated:
2018_05_21-PM-03_18_42
Last ObjectModification:
2018_05_19-AM-08_10_23
Theory : rings_1
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