Nuprl Lemma : p-mul-int-cancelation-1
∀[p:{2...}]. ∀[k:ℕ]. ∀[a,b:p-adics(p)]. ((p^k(p) * a = p^k(p) * b ∈ p-adics(p)) ⇒ (a = b ∈ p-adics(p)))
Proof
Definitions occuring in Statement :
p-int: k(p),
p-mul: x * y,
p-adics: p-adics(p),
exp: i^n,
int_upper: {i...},
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
int_upper: {i...},
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
p-int: k(p),
p-mul: x * y,
int_seg: {i..j-},
p-adics: p-adics(p),
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
eqmod: a ≡ b mod m,
divides: b | a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
lelt: i ≤ j < k
Lemmas referenced :
p-adics-equal,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
nat_plus_wf,
p-mul_wf,
p-int_wf,
exp_wf2,
p-adics_wf,
nat_wf,
int_upper_wf,
nat_plus_subtype_nat,
p-reduce_wf,
eqmod_functionality_wrt_eqmod,
eqmod_transitivity,
p-reduce-eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening,
nat_plus_properties,
nat_properties,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
p-adic-property,
decidable__le,
le_wf,
eqmod_inversion,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
exp-positive,
exp_add,
subtract_wf,
mul_cancel_in_eq,
exp_wf3,
subtype_rel_sets,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
exp_wf_nat_plus,
int_seg_properties,
decidable__equal_int,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality_alt,
because_Cache,
setElimination,
rename,
productElimination,
natural_numberEquality,
hypothesisEquality,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
isectElimination,
sqequalRule,
applyEquality,
universeIsType,
equalityIsType1,
inhabitedIsType,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
multiplyEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
intEquality,
instantiate,
cumulativity,
closedConclusion,
equalityIsType4,
baseApply,
baseClosed,
setIsType
Latex:
\mforall{}[p:\{2...\}]. \mforall{}[k:\mBbbN{}]. \mforall{}[a,b:p-adics(p)]. ((p\^{}k(p) * a = p\^{}k(p) * b) {}\mRightarrow{} (a = b))
Date html generated:
2019_10_15-AM-10_35_20
Last ObjectModification:
2018_10_15-PM-08_54_21
Theory : rings_1
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