Nuprl Lemma : p-int-injection
∀[p:{2...}]. Inj(ℤ;p-adics(p);λk.k(p))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-adics: p-adics(p),
inject: Inj(A;B;f),
int_upper: {i...},
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
int_upper: {i...},
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_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,
nat: ℕ,
exists: ∃x:A. B[x],
guard: {T},
so_lambda: λ2x.t[x],
p-adics: p-adics(p),
int_seg: {i..j-},
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
lelt: i ≤ j < k,
cand: A c∧ B,
less_than: a < b
Lemmas referenced :
equal_wf,
p-adics_wf,
p-int_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
int_upper_wf,
decidable__le,
p-int-eventually-constant,
le_wf,
all_wf,
less_than_transitivity1,
int_seg_wf,
exp_wf2,
upper_subtype_nat,
sq_stable__le,
le_weakening2,
or_wf,
equal-wf-T-base,
int_subtype_base,
p-minus-int-eventually,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
nat_plus_wf,
minus-minus,
nat_plus_properties,
decidable__equal_int,
subtract-is-int-iff,
intformeq_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
imax_wf,
imax_ub,
subtype_rel_self,
iff_weakening_equal,
imax_nat_plus,
imax_nat,
nat_plus_subtype_nat,
nat_wf,
nat_properties,
int_seg_properties,
add-is-int-iff,
add_nat_plus,
add_nat_wf,
exp-increasing
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
productElimination,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
dependent_pairFormation,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
minusEquality,
approximateComputation,
int_eqEquality,
inlFormation,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
universeEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
instantiate,
applyLambdaEquality
Latex:
\mforall{}[p:\{2...\}]. Inj(\mBbbZ{};p-adics(p);\mlambda{}k.k(p))
Date html generated:
2018_05_21-PM-03_19_26
Last ObjectModification:
2018_05_19-AM-08_11_21
Theory : rings_1
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