Nuprl Lemma : pa-mul_functionality
∀[p,q:{2...}]. ∀[x1,y1,x2,y2:basic-padic(p)].
(pa-mul(p;x1;y1) = pa-mul(q;x2;y2) ∈ padic(p)) supposing ((p = q ∈ ℤ) and bpa-equiv(p;x1;x2) and bpa-equiv(p;y1;y2))
Proof
Definitions occuring in Statement :
pa-mul: pa-mul(p;x;y),
padic: padic(p),
bpa-equiv: bpa-equiv(p;x;y),
basic-padic: basic-padic(p),
int_upper: {i...},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
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,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
pa-mul: pa-mul(p;x;y),
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
basic-padic: basic-padic(p),
bpa-equiv: bpa-equiv(p;x;y),
bpa-mul: bpa-mul(p;x;y),
top: Top,
nat: ℕ,
squash: ↓T,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
p-adics: p-adics(p),
so_lambda: λ2x.t[x],
subtract: n - m,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_apply: x[s]
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
equal-padics,
pa-mul_wf,
bpa-equiv-iff-norm,
bpa-mul_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
equal_wf,
bpa-equiv_wf,
basic-padic_wf,
int_upper_wf,
p-adics_wf,
p-mul_wf,
exp_wf2,
squash_wf,
true_wf,
nat_plus_wf,
p-int_wf,
exp_add,
subtype_rel_self,
p-mul-int,
iff_weakening_equal,
p-adic-property,
nat_plus_properties,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
p-mul-comm,
all_wf,
eqmod_wf,
nat_plus_subtype_nat,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
int_seg_wf,
int_seg_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
p-mul-assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
hypothesisEquality,
productElimination,
dependent_set_memberEquality,
setElimination,
rename,
because_Cache,
natural_numberEquality,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
sqequalRule,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
addEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
minusEquality,
applyLambdaEquality
Latex:
\mforall{}[p,q:\{2...\}]. \mforall{}[x1,y1,x2,y2:basic-padic(p)].
(pa-mul(p;x1;y1) = pa-mul(q;x2;y2)) supposing
((p = q) and
bpa-equiv(p;x1;x2) and
bpa-equiv(p;y1;y2))
Date html generated:
2018_05_21-PM-03_27_00
Last ObjectModification:
2018_05_19-AM-08_24_23
Theory : rings_1
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