Nuprl Lemma : p-mul_wf
∀[p:ℕ+]. ∀[x,y:p-adics(p)]. (x * y ∈ p-adics(p))
Proof
Definitions occuring in Statement :
p-mul: x * y,
p-adics: p-adics(p),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
p-mul: x * y,
p-adics: p-adics(p),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
lelt: i ≤ j < k,
so_apply: x[s],
int_upper: {i...}
Lemmas referenced :
p-reduce_wf,
nat_plus_subtype_nat,
int_seg_wf,
exp_wf2,
nat_plus_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,
int_seg_properties,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
p-adics_wf,
p-reduce-eqmod-exp,
eqmod_functionality_wrt_eqmod,
eqmod_weakening,
p-reduce-eqmod,
multiply_functionality_wrt_eqmod,
eqmod_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
lambdaEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
multiplyEquality,
natural_numberEquality,
because_Cache,
lambdaFormation,
addEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
minusEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
axiomEquality,
functionExtensionality
Latex:
\mforall{}[p:\mBbbN{}\msupplus{}]. \mforall{}[x,y:p-adics(p)]. (x * y \mmember{} p-adics(p))
Date html generated:
2018_05_21-PM-03_18_23
Last ObjectModification:
2018_05_19-AM-08_09_51
Theory : rings_1
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